An introduction to non-abelian discrete symmetries for particle physicists

Lecture Notes in Physics Volume 858

Founding Editors W. Beiglböck J. Ehlers K. Hepp H. Weidenmüller Editorial Board B.-G. Englert, Singapore, Singapore U. Frisch, Nice, France P. Hänggi, Augsburg, Germany W. Hillebrandt, Garching, Germany M. Hjort-Jensen, Oslo, Norway R. A. L. Jones, Sheffield, UK H. von Löhneysen, Karlsruhe, Germany M. S. Longair, Cambridge, UK M. L. Mangano, Geneva, Switzerland J.-F. Pinton, Lyon, France J.-M. Raimond, Paris, France A. Rubio, Donostia, San Sebastian, Spain M. Salmhofer, Heidelberg, Germany D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Garching, Germany

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The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching—quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany [email protected]

Hajime Ishimori r Tatsuo Kobayashi r Hiroshi Ohki r Hiroshi Okada r Yusuke Shimizu Morimitsu Tanimoto

An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists

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Hajime Ishimori Department of Physics Kyoto University Kyoto, Japan Tatsuo Kobayashi Department of Physics Kyoto University Kyoto, Japan Hiroshi Ohki Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMJ) Nagoya University Nagoya, Japan

Hiroshi Okada School of Physics Korean Institute for Advanced Study Seoul, Korea Yusuke Shimizu Department of Physics Niigata University Niigata, Japan Morimitsu Tanimoto Department of Physics Niigata University Niigata, Japan

ISSN 0075-8450 ISSN 1616-6361 (electronic) Lecture Notes in Physics ISBN 978-3-642-30804-8 ISBN 978-3-642-30805-5 (eBook) DOI 10.1007/978-3-642-30805-5 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012944594 © Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The purpose of these lecture notes is to introduce the basic framework of nonAbelian discrete symmetries, and to present some important applications in particle physics. Discrete non-Abelian groups have in fact played an important role in particle physics. However, they may not be so familiar to particle physicists as continuous non-Abelian symmetries. These lecture notes are written for particle physicists and differ in this respect from standard books on group theory. However, preliminary knowledge of group theory is not required to understand the non-Abelian discrete symmetries. We hope our lecture notes will serve as a handbook for serious learners, and also as a helpful reference book for experts, as well perhaps as triggering future research. It is pleasure to acknowledge fruitful discussions with H. Abe, T. Araki, K.S. Choi, Y. Daikoku, K. Hashimoto, J. Kubo, H.P. Nilles, F. Ploger, S. Raby, S. Ramos-Sanchez, M. Ratz, and P.K.S. Vaudrevange. Hajime Ishimori Tatsuo Kobayashi Hiroshi Ohki Hiroshi Okada Yusuke Shimizu Morimitsu Tanimoto

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

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Basics of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SN . . . . 3.1 S3 . . 3.1.1 3.1.2 3.1.3 3.2 S4 . . 3.2.1 3.2.2 3.2.3 References

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AN . . . . 4.1 A4 . 4.2 A5 . 4.2.1 4.2.2 4.2.3 References

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T 5.1 5.2 5.3

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DN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 DN with N Even . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Conjugacy Classes . . . . . . . . . . . . . . . . . . . . . .

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6.1.2 Characters and Representations 6.1.3 Tensor Products . . . . . . . . 6.2 DN with N Odd . . . . . . . . . . . . 6.2.1 Conjugacy Classes . . . . . . . 6.2.2 Characters and Representations 6.2.3 Tensor Products . . . . . . . . 6.3 D4 . . . . . . . . . . . . . . . . . . . 6.4 D5 . . . . . . . . . . . . . . . . . . .

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QN . . . . . . . . . . . . . . . . . . . . . . 7.1 QN with N = 4n . . . . . . . . . . . . 7.1.1 Conjugacy Classes . . . . . . . 7.1.2 Characters and Representations 7.1.3 Tensor Products . . . . . . . . 7.2 QN with N = 4n + 2 . . . . . . . . . 7.2.1 Conjugacy Classes . . . . . . . 7.2.2 Characters and Representations 7.2.3 Tensor Products . . . . . . . . 7.3 Q4 . . . . . . . . . . . . . . . . . . . 7.4 Q6 . . . . . . . . . . . . . . . . . . .

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QD2N . . . . . . . . . . . . . . . . . . . . 8.1 Generic Aspects . . . . . . . . . . . . 8.1.1 Conjugacy Classes . . . . . . . 8.1.2 Characters and Representations 8.1.3 Tensor Products . . . . . . . . 8.2 QD16 . . . . . . . . . . . . . . . . . .

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Σ(2N 2 ) . . . . . . . . . . . . . . . . . . . 9.1 Generic Aspects . . . . . . . . . . . . 9.1.1 Conjugacy Classes . . . . . . . 9.1.2 Characters and Representations 9.1.3 Tensor Products . . . . . . . . 9.2 Σ(18) . . . . . . . . . . . . . . . . . 9.3 Σ(32) . . . . . . . . . . . . . . . . . 9.4 Σ(50) . . . . . . . . . . . . . . . . .

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Δ(3N 2 ) . . . . . . . . . . . . . . . . . . . 10.1 Δ(3N 2 ) with N/3 = Integer . . . . . . 10.1.1 Conjugacy Classes . . . . . . . 10.1.2 Characters and Representations 10.1.3 Tensor Products . . . . . . . . 10.2 Δ(3N 2 ) with N/3 Integer . . . . . . . 10.2.1 Conjugacy Classes . . . . . . . 10.2.2 Characters and Representations 10.2.3 Tensor Products . . . . . . . . 10.3 Δ(27) . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

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11

TN . . . . . . . . . . . . . . . . . . . . . . 11.1 Generic Aspects . . . . . . . . . . . . 11.1.1 Conjugacy Classes . . . . . . . 11.1.2 Characters and Representations 11.1.3 Tensor Products . . . . . . . . 11.2 T7 . . . . . . . . . . . . . . . . . . . . 11.3 T13 . . . . . . . . . . . . . . . . . . . 11.4 T19 . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

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Σ(3N 3 ) . . . . . . . . . . . . . . . . . . . 12.1 Generic Aspects . . . . . . . . . . . . 12.1.1 Conjugacy Classes . . . . . . . 12.1.2 Characters and Representations 12.1.3 Tensor Products . . . . . . . . 12.2 Σ(81) . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

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Δ(6N 2 ) . . . . . . . . . . . . . . . . . . . 13.1 Δ(6N 2 ) with N/3 = Integer . . . . . . 13.1.1 Conjugacy Classes . . . . . . . 13.1.2 Characters and Representations 13.1.3 Tensor Products . . . . . . . . 13.2 Δ(6N 2 ) with N/3 Integer . . . . . . . 13.2.1 Conjugacy Classes . . . . . . . 13.2.2 Characters and Representations 13.2.3 Tensor Products . . . . . . . . 13.3 Δ(54) . . . . . . . . . . . . . . . . . 13.3.1 Conjugacy Classes . . . . . . . 13.3.2 Characters and Representations 13.3.3 Tensor Products . . . . . . . . References . . . . . . . . . . . . . . . . . .

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123 123 123 126 128 131 131 133 134 138 138 139 141 145

14 Subgroups and Decompositions of Multiplets 14.1 S3 . . . . . . . . . . . . . . . . . . . . . . 14.1.1 S3 → Z3 . . . . . . . . . . . . . . 14.1.2 S3 → Z2 . . . . . . . . . . . . . . 14.2 S4 . . . . . . . . . . . . . . . . . . . . . . 14.2.1 S4 → S3 . . . . . . . . . . . . . . 14.2.2 S4 → A4 . . . . . . . . . . . . . . 14.2.3 S4 → Σ(8) . . . . . . . . . . . . . 14.3 A4 . . . . . . . . . . . . . . . . . . . . . 14.3.1 A4 → Z3 . . . . . . . . . . . . . . 14.3.2 A4 → Z2 × Z2 . . . . . . . . . . . 14.4 A5 . . . . . . . . . . . . . . . . . . . . . 14.4.1 A5 → A4 . . . . . . . . . . . . . .

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14.4.2 A5 → D5 . . . . . . . 14.4.3 A5 → S3  D3 . . . . 14.5 T  . . . . . . . . . . . . . . . 14.5.1 T  → Z6 . . . . . . . 14.5.2 T  → Z4 . . . . . . . 14.5.3 T  → Q4 . . . . . . . 14.6 General DN . . . . . . . . . . 14.6.1 DN → Z2 . . . . . . 14.6.2 DN → ZN . . . . . . 14.6.3 DN → DM . . . . . . 14.7 D4 . . . . . . . . . . . . . . . 14.7.1 D4 → Z4 . . . . . . . 14.7.2 D4 → Z2 × Z2 . . . . 14.7.3 D4 → Z2 . . . . . . . 14.8 General QN . . . . . . . . . . 14.8.1 QN → Z4 . . . . . . 14.8.2 QN → ZN . . . . . . 14.8.3 QN → QM . . . . . . 14.9 Q4 . . . . . . . . . . . . . . . 14.9.1 Q4 → Z4 . . . . . . . 14.10 QD2N . . . . . . . . . . . . . 14.10.1 QD2N → Z2 . . . . . 14.10.2 QD2N → ZN . . . . . 14.10.3 QD2N → DN/2 . . . . 14.11 General Σ(2N 2 ) . . . . . . . 14.11.1 Σ(2N 2 ) → Z2N . . . 14.11.2 Σ(2N 2 ) → ZN × ZN 14.11.3 Σ(2N 2 ) → DN . . . 14.11.4 Σ(2N 2 ) → QN . . . 14.11.5 Σ(2N 2 ) → Σ(2M 2 ) . 14.12 Σ(32) . . . . . . . . . . . . . 14.13 General Δ(3N 2 ) . . . . . . . 14.13.1 Δ(3N 2 ) → Z3 . . . . 14.13.2 Δ(3N 2 ) → ZN × ZN 14.13.3 Δ(3N 2 ) → TN . . . . 14.13.4 Δ(3N 2 ) → Δ(3M 2 ) . 14.14 Δ(27) . . . . . . . . . . . . . 14.14.1 Δ(27) → Z3 . . . . . 14.14.2 Δ(27) → Z3 × Z3 . . 14.15 General TN . . . . . . . . . . 14.15.1 TN → Z3 . . . . . . . 14.15.2 TN → ZN . . . . . . 14.16 T7 . . . . . . . . . . . . . . . 14.16.1 T7 → Z3 . . . . . . . 14.16.2 T7 → Z7 . . . . . . .

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14.17 General Σ(3N 3 ) . . . . . . . . . . 14.17.1 Σ(3N 2 ) → ZN × ZN × ZN 14.17.2 Σ(3N 3 ) → Δ(3N 2 ) . . . . 14.17.3 Σ(3N 3 ) → Σ(3M 3 ) . . . . 14.18 Σ(81) . . . . . . . . . . . . . . . . 14.18.1 Σ(81) → Z3 × Z3 × Z3 . . 14.18.2 Σ(81) → Δ(27) . . . . . . 14.19 General Δ(6N 2 ) . . . . . . . . . . 14.19.1 Δ(6N 2 ) → Σ(2N 2 ) . . . . 14.19.2 Δ(6N 2 ) → Δ(3N 2 ) . . . . 14.19.3 Δ(6N 2 ) → Δ(6M 2 ) . . . . 14.20 Δ(54) . . . . . . . . . . . . . . . . 14.20.1 Δ(54) → S3 × Z3 . . . . . 14.20.2 Δ(54) → Σ(18) . . . . . . 14.20.3 Δ(54) → Δ(27) . . . . . .

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175 175 175 176 176 177 177 178 179 180 180 181 182 182 183

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185 185 189 189 190 190 191 192 193 194 194 195 196 197 198 199 200 201 202 203 203 204

16 Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models 16.1 Neutrino Flavor Mixing and Neutrino Mass Matrix . . . . . . 16.2 A4 Flavor Symmetry . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Realizing Tri-Bimaximal Mixing of Flavors . . . . . . 16.2.2 Breaking Tri-Bimaximal Mixing . . . . . . . . . . . . 16.3 S4 Flavor Model . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Alternative Flavor Mixing . . . . . . . . . . . . . . . . . . .

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205 205 207 207 209 211 219

15 Anomalies . . . . . . . . . . . . . . . . 15.1 Generic Aspects . . . . . . . . . . 15.2 Explicit Calculations . . . . . . . 15.2.1 S3 . . . . . . . . . . . . . 15.2.2 S4 . . . . . . . . . . . . . 15.2.3 A4 . . . . . . . . . . . . 15.2.4 A5 . . . . . . . . . . . . 15.2.5 T  . . . . . . . . . . . . . 15.2.6 DN (N Even) . . . . . . . 15.2.7 DN (N Odd) . . . . . . . 15.2.8 QN (N = 4n) . . . . . . . 15.2.9 QN (N = 4n + 2) . . . . 15.2.10 QD2N . . . . . . . . . . . 15.2.11 Σ(2N 2 ) . . . . . . . . . 15.2.12 Δ(3N 2 ) (N/3 = Integer) . 15.2.13 Δ(3N 2 ) (N/3 Integer) . . 15.2.14 TN . . . . . . . . . . . . 15.2.15 Σ(3N 3 ) . . . . . . . . . 15.2.16 Δ(6N 2 ) (N/3 = Integer) . 15.2.17 Δ(6N 2 ) (N/3 Integer) . . 15.3 Comments on Anomalies . . . . . References . . . . . . . . . . . . . . . .

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16.5 Comments on Other Applications . . . . . . . . . . . . . . . . . 222 16.6 Comment on Origins of Flavor Symmetries . . . . . . . . . . . . 223 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Appendix A Useful Theorems . . . . . . . . . . . . . . . . . . . . . . . . 229 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Appendix B Representations of S4 in Different Bases B.1 Basis I . . . . . . . . . . . . . . . . . . . . . B.2 Basis II . . . . . . . . . . . . . . . . . . . . B.3 Basis III . . . . . . . . . . . . . . . . . . . . B.4 Basis IV . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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237 237 238 240 242 244

Appendix C Representations of A4 in Different Bases C.1 Basis I . . . . . . . . . . . . . . . . . . . . . . C.2 Basis II . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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245 245 245 246

Appendix D Representations of A5 in Different Bases D.1 Basis I . . . . . . . . . . . . . . . . . . . . . . D.2 Basis II . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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247 247 253 259

Appendix E Representations of T  in Different Bases E.1 Basis I . . . . . . . . . . . . . . . . . . . . . E.2 Basis II . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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261 262 263 264

Appendix F Other Smaller Groups F.1 Z4  Z4 . . . . . . . . . . F.2 Z8  Z2 . . . . . . . . . . F.3 (Z2 × Z4 )  Z2 (I) . . . . F.4 (Z2 × Z4 )  Z2 (II) . . . . F.5 Z3  Z8 . . . . . . . . . . F.6 (Z6 × Z2 )  Z2 . . . . . . F.7 Z9  Z3 . . . . . . . . . . References . . . . . . . . . . . .

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Chapter 1

Introduction

These lecture notes aim to provide a pedagogical review of non-Abelian discrete groups and show some applications to physical issues. Symmetry constitutes a very important principle in physics. In particular, it has played an essential role in constructing the framework of particle physics. For example, continuous (and local) symmetries such as Lorentz, Poincaré, and gauge symmetries are crucial to understand several phenomena, such as the strong, weak, and electromagnetic interactions among particles. On the other hand, discrete symmetries such as C, P , and T are also vital concepts in particle physics. Abelian discrete symmetries, ZN , are also often imposed in order to control allowed couplings for particle physics, in particular model-building beyond the standard model. In addition to Abelian discrete symmetries, non-Abelian discrete symmetries have also been applied for model-building in particle physics, in particular to understand the three-generation flavor structure. There are many free parameters in the standard model, including its extension with neutrino mass terms. Most of them are Yukawa couplings of quarks and leptons to the Higgs boson. The quark and lepton sector is called the flavor sector. Flavor physics is a challenging aspect of the construction of the theory beyond the standard model. If a symmetry is imposed on the flavor sector, one can control the Yukawa couplings in the three generations, although the origin of the generations remains unknown. Therefore, quark masses and mixing angles have been studied from the standpoint of flavor symmetries. In addition, the discovery of neutrino masses and neutrino mixing [1, 2] has stimulated work on flavor symmetries. Experiments on neutrino oscillations are now going into a new phase of precise determination of mixing angles and mass squared differences [3–7]. In particular, the recent long baseline neutrino experiment T2K is reaching the last neutrino mixing angle, so called θ13 [8]. The Double Chooz collaboration has also reported indications of non-zero θ13 [9]. Reactor neutrino experiments, Reno and Daya Bay are also attempting to observe it. Global analyses of neutrino data indicate the special neutrino mixing pattern, which is called tribimaximal mixing for three flavors in the lepton sector [10–13]. These large mixing angles are completely different from the quark mixing ones. Therefore, it is very H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_1, © Springer-Verlag Berlin Heidelberg 2012

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Introduction

important to find a natural model that leads to these mixing patterns of quarks and leptons with good accuracy. Non-Abelian discrete symmetries are considered to be the most attractive choice for the flavor sector. Model builders have tried to derive experimental values of quark/lepton masses and mixing angles by assuming non-Abelian discrete flavor symmetries of quarks and leptons. In particular, lepton mixing has been intensively discussed in the context of non-Abelian discrete flavor symmetries, as seen, e.g., in the reviews [14, 15]. Particle physicists may be interested in the origin of the non-Abelian discrete symmetry for flavors. One of the most interesting is a higher dimensional spacetime symmetry. After it has been broken down to the 4D Poincaré symmetry through compactification, e.g., via orbifolding, a remnant symmetry appears in the flavor sector. This remnant symmetry is often a non-Abelian symmetry. Actually, it has been shown how the flavor symmetry A4 (or S4 ) can arise if the three fermion generations are taken to live on the fixed points of a specific 2D orbifold [16]. Further non-Abelian discrete symmetries can arise in a similar setup [17] (see also [18]). Superstring theory is a promising candidate for a unified theory including gravity. Certain string modes correspond to gauge bosons, quarks, leptons, Higgs bosons, and gravitons as well as their superpartners. Superstring theory predicts six extra dimensions. Certain classes of discrete symmetries can be derived from superstring theories. A combination among geometrical symmetries of a compact space and stringy selection rules for couplings enhances discrete flavor symmetries. For example, D4 and Δ(54) flavor symmetries can be obtained in heterotic orbifold models [19–21]. In addition to these flavor symmetries, the Δ(27) flavor symmetry can be derived from magnetized/intersecting D-brane models [22–24]. There is another possibility, namely that non-Abelian discrete groups may originate from the breaking of continuous (gauge) flavor symmetries [25–27]. Thus, a non-Abelian discrete symmetry can arise from the underlying theory, e.g., string theory or compactification via orbifolding. In addition, non-Abelian discrete symmetries are interesting tools for controlling flavor structure in model building using the bottom-up approach. Hence, non-Abelian flavor symmetries could provide a bridge between the low-energy physics and the underlying theory. It is thus quite important to understand the properties of non-Abelian groups for particle physics. Continuous non-Abelian groups are well-known, and of course there are several good reviews and books. On the other hand, discrete non-Abelian symmetries may not be so familiar to particle physicists as continuous non-Abelian symmetries. However, discrete non-Abelian symmetries have become important tools for model building, as discussed above, in particular in the context of flavor physics. The purpose of these lecture notes is therefore to provide a pedagogical review of non-Abelian discrete groups with particle phenomenology in mind, and to exhibit the group-theoretical aspects of many concrete groups explicitly, including, for example, representations and their tensor products [15, 28–34]. We present these aspects in detail for the groups SN [35–132], AN [133–243], T  [33, 244–263], DN [264–285], QN [286–300], QD2N , Σ(2N 2 ) [301], Δ(3N 2 ) [302–313], TN [302– 304, 312, 314–323], Σ(3N 3 ) [315, 324], and Δ(6N 2 ) [302–304, 312, 325–330].

References

3

We explain pedagogically how to derive conjugacy classes, characters, representations, and tensor products for these groups (with a finite number) when algebraic relations are given. Thus, it will be straightforward for readers to apply this to other groups. In applications to particle physics, the breaking patterns of discrete groups and decompositions of multiplets are often required to understand low energy phenomena. Such aspects are given in these notes. Symmetries at the tree level are not always symmetries in quantum theory. If symmetries are anomalous, breaking terms are induced by quantum effects. Such anomalies are important in applications for particle physics. Here, we study such anomalies for discrete symmetries [331–344] and show anomaly-free conditions explicitly for the above concrete groups. If flavor symmetries are stringy symmetries, these anomalies may also be controlled by string dynamics, i.e., anomaly cancellation. We also present flavor models with non-Abelian discrete symmetry as typical examples. One can see how to use the non-Abelian discrete symmetry for flavors. A lot of references are available to understand the model building here. On the other hand, discrete subgroups of SU(3) would also be interesting from the standpoint of phenomenological applications to flavor physics [345–349]. Here, most of them are shown for subgroups including doublets or triplets as the largest dimensional irreducible representations (for other groups see [29, 31, 34, 241, 350– 354]). The book is organized as follows. In Chap. 2, we summarize the basic grouptheoretical aspects used in subsequent chapters, and also present some examples to provide a more concrete understanding. Readers familiar with group theory can skip Chap. 2. In Chaps. 3 to 13, we present the non-Abelian discrete groups SN , AN , T  , DN , QN , QD2N , Σ(2N 2 ), Δ(3N 2 ), TN , Σ(3N 3 ), and Δ(6N 2 ). In each chapter, groups with specific values of N are also discussed for typical examples. Chapter 14 discusses the breaking patterns of the non-Abelian discrete groups. In Chap. 15, we review the anomalies of non-Abelian flavor symmetries, which is an important topic in particle physics, and exhibit the anomaly-free conditions explicitly for the above concrete groups. Chapter 16 presents typical flavor models with the non-Abelian discrete symmetries A4 and S4 . Appendix A gives some useful theorems on finite group theory, while Appendices B, C, D, and E provide the representation bases of S4 , A4 , A5 , and T  , which are different from those in Chaps. 3, 4, and 5. Appendix F presents other smaller groups in detail. Note Added in Proof

Finally, θ13 has been observed by Daya Bay [355] and Reno [356].

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316. Cao, Q.-H., Khalil, S., Ma, E., Okada, H.: Phys. Rev. Lett. 106, 131801 (2011). arXiv:1009.5415 [hep-ph] 317. Ma, E.: Mod. Phys. Lett. A 26, 377–385 (2011). arXiv:1101.4972 [hep-ph] 318. Cao, Q.-H., Khalil, S., Ma, E., Okada, H.: arXiv:1108.0570 [hep-ph] 319. Kajiyama, Y., Okada, H.: Nucl. Phys. B 848, 303–313 (2011). arXiv:1011.5753 [hep-ph] 320. Parattu, K.M., Wingerter, A.: Phys. Rev. D 84, 013011 (2011). arXiv:1012.2842 [hep-ph] 321. Ding, G.-J.: Nucl. Phys. B 853, 635–662 (2011). arXiv:1105.5879 [hep-ph] 322. Hartmann, C., Zee, A.: Nucl. Phys. B 853, 105–124 (2011). arXiv:1106.0333 [hep-ph] 323. Hartmann, C.: arXiv:1109.5143 [hep-ph] 324. Ishimori, H., Kobayashi, T.: arXiv:1201.3429 [hep-ph] 325. Escobar, J.A., Luhn, C.: J. Math. Phys. 50, 013524 (2009). arXiv:0809.0639 [hep-th] 326. Ishimori, H., Kobayashi, T., Okada, H., Shimizu, Y., Tanimoto, M.: J. High Energy Phys. 0904, 011 (2009). arXiv:0811.4683 [hep-ph] 327. Ishimori, H., Kobayashi, T., Okada, H., Shimizu, Y., Tanimoto, M.: J. High Energy Phys. 0912, 054 (2009). arXiv:0907.2006 [hep-ph] 328. Escobar, J.A.: arXiv:1102.1649 [hep-ph] 329. Varzielas, I.d.M., Emmanuel-Costa, D.: arXiv:1106.5477 [hep-ph] 330. Toorop, R.d.A., Feruglio, F., Hagedorn, C.: Phys. Lett. B 703, 447–451 (2011). arXiv:1107.3486 [hep-ph] 331. Krauss, L.M., Wilczek, F.: Phys. Rev. Lett. 62, 1221 (1989) 332. Ibáñez, L.E., Ross, G.G.: Phys. Lett. B 260, 291–295 (1991) 333. Banks, T., Dine, M.: Phys. Rev. D 45, 1424–1427 (1992). arXiv:hep-th/9109045 334. Dine, M., Graesser, M.: J. High Energy Phys. 01, 038 (2005). arXiv:hep-th/0409209 335. Csaki, C., Murayama, H.: Nucl. Phys. B 515, 114–162 (1998). arXiv:hep-th/9710105 336. Ibáñez, L.E., Ross, G.G.: Nucl. Phys. B 368, 3–37 (1992) 337. Ibáñez, L.E.: Nucl. Phys. B 398, 301–318 (1993). arXiv:hep-ph/9210211 338. Babu, K.S., Gogoladze, I., Wang, K.: Nucl. Phys. B 660, 322–342 (2003). arXiv:hep-ph/0212245 339. Dreiner, H.K., Luhn, C., Thormeier, M.: Phys. Rev. D 73, 075007 (2006). arXiv:hep-ph/0512163 340. Araki, T.: Prog. Theor. Phys. 117, 1119–1138 (2007). arXiv:hep-ph/0612306 341. Araki, T., Choi, K.S., Kobayashi, T., Kubo, J., Ohki, H.: Phys. Rev. D 76, 066006 (2007). arXiv:0705.3075 [hep-ph] 342. Araki, T., Kobayashi, T., Kubo, J., Ramos-Sanchez, S., Ratz, M., Vaudrevange, P.K.S.: Nucl. Phys. B 805, 124 (2008). arXiv:0805.0207 [hep-th] 343. Luhn, C., Ramond, P.: J. High Energy Phys. 0807, 085 (2008). arXiv:0805.1736 [hep-ph] 344. Luhn, C.: Phys. Lett. B 670, 390 (2009). arXiv:0807.1749 [hep-ph] 345. Merle, A., Zwicky, R.: arXiv:1110.4891 [hep-ph] 346. Grimus, W., Ludl, P.O.: J. Phys. A 43, 445209 (2010). arXiv:1006.0098 [hep-ph] 347. Ludl, P.O.: J. Phys. A 43, 395204 (2010). arXiv:1006.1479 [math-ph] 348. Ludl, P.O.: J. Phys. A 44, 255204 (2011). arXiv:1101.2308 [math-ph] 349. Luhn, C.: J. High Energy Phys. 1103, 108 (2011). arXiv:1101.2417 [hep-ph] 350. King, S.F., Luhn, C.: Nucl. Phys. B 820, 269 (2009). arXiv:0905.1686 [hep-ph] 351. King, S.F., Luhn, C.: Nucl. Phys. B 832, 414 (2010). arXiv:0912.1344 [hep-ph] 352. Everett, L.L., Stuart, A.J.: Phys. Lett. B 698, 131–139 (2011). arXiv:1011.4928 [hep-ph] 353. Hashimoto, K., Okada, H.: arXiv:1110.3640 [hep-ph] 354. Chen, C.-S., Kephart, T.W., Yuan, T.-C.: arXiv:1110.6233 [hep-ph] 355. An, F.P., et al. (Daya Bay Collaboration): Phys. Rev. Lett. 108, 171803 (2012). arXiv: 1203.1669 [hep-ex] 356. Ahn, J.K., et al. (Reno Collaboration): Phys. Rev. Lett. 108, 191802 (2012). arXiv:1204.0626 [hep-ex]

Chapter 2

Basics of Finite Groups

We start by introducing the basics of group theory, considering in particular finite groups. For pedagogical purposes, we shall use several theorems without proof, although proofs of useful theorems are given in Appendix A. (See also, e.g., [1–6].) On the other hand, we shall present several examples in order to obtain a clear understanding of these basic theorems. A group G is a set with a product satisfying the following properties: 1. Closure If a and b are elements of the group G, then c = ab is also an element of G. 2. Associativity (ab)c = a(bc) for all a, b, c ∈ G. 3. Identity The group G includes an identity element e, which satisfies ae = ea = a for any element a ∈ G. 4. Inverse The group G includes an inverse element a −1 for any element a ∈ G, such that aa −1 = a −1 a = e. Let us present some simple examples. Example (Cyclic Group ZN ) Discrete rotations of a complex plane form a group. Let us denote the exp[2π i/N ] rotation by a. Then the exp[2π im/N ] rotation for m = integer can be written a m . The multiplication rule is defined such that a m a n = a m+n . The operator a N corresponds to the identity, a N = e, and the inverse of a m is obtained as a N−m . Thus, the set 

e, a, a 2 , . . . , a N −1



(2.1)

forms a group. Its closure and associativity should be obvious. This group is called the cyclic group ZN . H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_2, © Springer-Verlag Berlin Heidelberg 2012

13

14

2

Basics of Finite Groups

Example (S3 and SN ) All possible permutations among three objects, (x1 , x2 , x3 ), form a group denoted by S3 . There are six permutations: e : (x1 , x2 , x3 ) → (x1 , x2 , x3 ), a1 : (x1 , x2 , x3 ) → (x2 , x1 , x3 ), a2 : (x1 , x2 , x3 ) → (x3 , x2 , x1 ), a3 : (x1 , x2 , x3 ) → (x1 , x3 , x2 ), a4 : (x1 , x2 , x3 ) → (x3 , x1 , x2 ), a5 : (x1 , x2 , x3 ) → (x2 , x3 , x1 ).

(2.2)

The element e is clearly the identity. Their products form a closed algebra, e.g., a1 a2 : (x1 , x2 , x3 ) → (x2 , x3 , x1 ), a2 a1 : (x1 , x2 , x3 ) → (x3 , x1 , x2 ),

(2.3)

a4 a2 : (x1 , x2 , x3 ) → (x1 , x3 , x2 ), whence a1 a 2 = a 5 ,

a2 a 1 = a 4 ,

a4 a2 = a2 a1 a2 = a3 .

(2.4)

It is straightforward to check the closure rule for other products, as well as associativity and the presence of an inverse for each element. Using the multiplication rules, one can write all six elements in terms of two proper elements and their products. For example, by defining a1 = a, a2 = b, all elements are written as {e, a, b, ab, ba, bab}.

(2.5)

Note that aba = bab. The group S3 is the symmetry group of an equilateral triangle, as shown in Fig. 2.1. The elements a and ab correspond to a reflection and the 2π/3 rotation, respectively. Similarly, all possible permutations among N objects xi with i = 1, . . . , N , (x1 , . . . , xN ) → (xi1 , . . . , xiN ),

(2.6)

form a group. This is denoted by SN and contains N ! elements. It is often called the symmetric group. The order of a group G is the number of elements in G. Obviously, the order of a finite group is finite. For example, the order of the group ZN is N , while the order of the group SN is N !. A group G is said to be Abelian if all its elements commute with each other, i.e., ab = ba for any elements a and b in G. If not all pairs of elements satisfy commutativity, the group is said to be non-Abelian. The group ZN is Abelian, but S3 and SN (N ≥ 3) are non-Abelian. For example, for S3 , we see that a1 a2 = a2 a1 in the above notation (2.2). If a subset H of a group G is also a group, H is said to be a subgroup of G. The order of the subgroup H is always a divisor of the order of G. This is Lagrange’s theorem (see Appendix A). If a subgroup N of G satisfies g −1 Ng = N for any element g ∈ G, the subgroup N is called a normal subgroup or an invariant subgroup.

2 Basics of Finite Groups

15

Fig. 2.1 The S3 symmetry of an equilateral triangle

Any subgroup H and normal subgroup N of G satisfy H N = N H , where H N denotes {hi nj |hi ∈ H, nj ∈ N },

(2.7)

and N H has a similar meaning. Furthermore, H N is a subgroup of G. Example For example, the three elements {e, ab, ba} form a subgroup of S3 . Indeed, these elements correspond to even permutations, while the other elements, {a, b, bab}, correspond to odd permutations. This subgroup is nothing but the Z3 group, because (ab)2 = ba and (ab)3 = e. Lagrange’s theorem implies that the order of any subgroup of S3 must be equal to 1, 2, or 3, because the order of S3 is 6 (= 3 × 2). The subgroup of order 3 corresponds to the above Z3 . In addition, the S3 group includes three subgroups of order 2, viz., {e, a}, {e, b}, and {e, bab}. These subgroups are Z2 groups. Furthermore, it can be shown that the above Z3 is a normal subgroup of S3 . When a h = e for an element a ∈ G and h is the smallest positive integer for which this is so, the number h is called the order of a. The elements {e, a, a 2 , . . . , a h−1 } form a subgroup, which is the Abelian group Zh of order h. The elements g −1 ag for g ∈ G are called elements conjugate to the element a. The set containing all elements conjugate to an element a of G, i.e., {g −1 ag, ∀g ∈ G}, is called a conjugacy class. All elements in a conjugacy class have the same order since  h     gag −1 = ga g −1 g a g −1 g · · · ag −1 = ga h g −1 = geg −1 = e. (2.8) The conjugacy class containing the identity e consists of the single element e. Example All the elements of S3 are classified into three conjugacy classes: C1 : {e},

C2 : {ab, ba},

C3 : {a, b, bab}.

(2.9)

Here, the subscript n of Cn denotes the number of elements in the conjugacy class Cn .

16

2

Basics of Finite Groups

We consider two groups, G and G , and a map f of G into G . This map is homomorphic if and only if it preserves the multiplicative structure, that is, f (a)f (b) = f (ab),

(2.10)

for all a, b ∈ G. Furthermore, the map is isomorphic when it is a one-to-one correspondence. A representation D of G is a homomorphic map of elements of G onto matrices D(g) for g ∈ G. The representation matrices then satisfy D(a)D(b) = D(c) if ab = c for a, b, c ∈ G. The vector space V on which the representation matrices act is called a representation space, with D(g)ij vj , (j = 1, . . . , n), for v ∈ V with components vj relative to some basis {e1 , . . . , en }. The dimension n of the vector space V is called the dimension of the representation. A subspace in the representation space is said to be an invariant subspace if, for any vector v in the subspace and any element g ∈ G, D(g)ij vj also corresponds to a vector in the same subspace. If there is an invariant subspace, such a representation is said to be reducible. In contrast, a representation is irreducible if it has no invariant subspace. In particular, a representation is said to be completely reducible if, for every g ∈ G, D(g) can be written in the following block diagonal form: ⎞ ⎛ 0 D1 (g) ⎟ ⎜ 0 D2 (g) ⎟ ⎜ (2.11) ⎟, ⎜ . .. ⎠ ⎝ Dr (g) where each Dα (g) is irreducible for α = 1, . . . , r. We then say that the reducible representation D(g) is the direct sum of the Dα (g): r

(2.12)

Dα (g).

α=1

Every (reducible) representation of a finite group is completely reducible. Furthermore, every representation of a finite group is equivalent to a unitary representation (see Appendix A). The simplest (irreducible) representation is just D(g) = 1 for all elements g, that is, a trivial 1D representation or singlet. The matrix representations satisfy the following orthogonality relation:

  NG Dα (g)i Dβ g −1 mj = δαβ δij δm , (2.13) dα g∈G

where NG is the order of G and dα is the dimension of Dα (g) for each α (see Appendix A). The character χD (g) of a representation D(g) is the trace of the representation matrix: χD (g) = tr D(g) =

dα

i=1

D(g)ii .

(2.14)

2 Basics of Finite Groups

17

The elements conjugate to a have the same character because of the following property of the trace:       tr D g −1 ag = tr D g −1 D(a)D(g) = tr D(a). (2.15) That is, the characters are constant in a conjugacy class. The characters satisfy the following orthogonality relation:

χDα (g)∗ χDβ (g) = NG δαβ , (2.16) g∈G

where NG denotes the order of a group G (see Appendix A). That is, the characters of different irreducible representations are orthogonal and different from each other. Furthermore, it can be shown that the number of irreducible representations must be equal to the number of conjugacy classes (see Appendix A). In addition, they satisfy the following orthogonality relation:

NG χDα (gi )∗ χDα (gj ) = δC C , (2.17) ni i j α where Ci denotes the conjugacy class of gi and ni denotes the number of elements in the conjugacy class Ci (see Appendix A). The right-hand side is equal to NG /ni if gi and gj belong to the same conjugacy class, and otherwise it must vanish. A trivial singlet, D(g) = 1 for any g ∈ G, must always be included. Thus, the corresponding character satisfies χ1 (g) = 1 for any g ∈ G. Suppose that there are mn n-dimensional irreducible representations, that is, with D(g) represented by (n × n) matrices. The identity e is always represented by the (n × n) identity matrix. Clearly, the character χDα (C1 ) for the conjugacy class C1 = {e} is just χDα (C1 ) = n for an n-dimensional representation. The orthogonality relation (2.17) then requires

 2 χα (C1 ) = mn n2 = m1 + 4m2 + 9m3 + · · · = NG , (2.18) α

n

where mn ≥ 0. Furthermore, mn must satisfy

mn = number of conjugacy classes,

(2.19)

n

because the number of irreducible representations is equal to the number of conjugacy classes. Equations (2.18) and (2.19), together with (2.16) and (2.17), will often be used in the following sections to determine characters. Example Let us study the irreducible representations of S3 . The number of irreducible representations must be equal to three, because there are three conjugacy classes. We assume that there are mn n-dimensional representations, that is, with  D(g) represented by (n × n) matrices. Here, mn must satisfy n mn = 3. Furthermore, the orthogonality relation (2.18) requires

 2 χα (C1 ) = mn n2 = m1 + 4m2 + 9m3 + · · · = 6, (2.20) α

n

18

2

Table 2.1 Characters of S3 representations

Basics of Finite Groups

h

χ1

χ1 

χ2

C1

1

1

1

2

C2

3

1

1

−1

C3

2

1

−1

0

where mn ≥ 0. This equation has only two possible solutions, (m1 , m2 ) = (2, 1) and (6, 0), but only the former (m1 , m2 ) = (2, 1) satisfies m1 + m2 = 3. Thus, irreducible representations of S3 include two singlets 1 and 1 , and a doublet 2. We denote their characters by χ1 (g), χ1 (g), and χ2 (g), respectively. Clearly, χ1 (C1 ) = χ1 (C1 ) = 1 and χ2 (C1 ) = 2. Furthermore, one of the singlet representations corresponds to a trivial singlet, that is, χ1 (C2 ) = χ1 (C3 ) = 1. The characters, which are not fixed at this stage, are χ1 (C2 ), χ1 (C3 ), χ2 (C2 ), and χ2 (C3 ). Now let us determine them. For a non-trivial singlet 1 , the representation matrices are nothing but the characters, χ1 (C2 ) and χ1 (C3 ). They must satisfy 3  2  χ1 (C3 ) = 1. (2.21) χ1 (C2 ) = 1, Thus, χ1 (C2 ) is one of 1, ω, and ω2 , where ω = exp[2πi/3], and χ1 (C3 ) is 1 or −1. On top of that, the orthogonality relation (2.16) requires

χ1 (g)χ1 (g) = 1 + 2χ1 (C2 ) + 3χ1 (C3 ) = 0. (2.22) g

Its unique solution is χ1 (C2 ) = 1 and χ1 (C3 ) = −1. Furthermore, the orthogonality relations (2.16) and (2.17) require

χ1 (g)χ2 (g) = 2 + 2χ2 (C2 ) + 3χ2 (C3 ) = 0, (2.23)

g

χα (C1 )∗ χα (C2 ) = 1 + χ1 (C2 ) + 2χ2 (C2 ) = 0.

(2.24)

α

Their solution is χ2 (C2 ) = −1 and χ2 (C3 ) = 0. These results are shown in Table 2.1. Next, we figure out the representation matrices D(g) of S3 using the character Table 2.1. For singlets, their characters are nothing but representation matrices. Let us consider representation matrices D(g) for the doublet, where D(g) are (2 × 2) unitary matrices. Obviously, D2 (e) is the (2 × 2) identity matrix. Since χ2 (C3 ) = 0, we can diagonalize one element of the conjugacy class C3 . Here we choose, e.g., a in C3 , as the diagonal element:   1 0 a= . (2.25) 0 −1 The other elements in C3 , as well as those in C2 , are non-diagonal matrices. Recalling that b2 = e, we can write     cos θ sin θ cos 2θ sin 2θ b= , bab = . (2.26) sin θ − cos θ sin 2θ − cos 2θ

2 Basics of Finite Groups

19

Then, we can write elements in C2 as   cos θ sin θ ab = , − sin θ cos θ

 ba =

cos θ sin θ

− sin θ cos θ

 .

(2.27)

Recall that the trace of elements in C2 is equal to −1, whence cos θ = −1/2, that is, θ = 2π/3, 4π/3. When we choose θ = 4π/3, we obtain the matrix representation of S3 as ⎛ √ ⎞     3 1 − − 1 0 1 0 2 2 ⎠ , e= , a= , b=⎝ √ 0 1 0 −1 1 − 23 2 ⎛ ⎛ ⎛ √ ⎞ √ ⎞ √ ⎞ 3 3 3 1 1 1 −2 − 2 −2 − 2 ⎠ 2 2 ⎠ ⎠, ab = ⎝ √ ba = ⎝ √ , bab = ⎝ √ . 3 3 3 1 1 1 − − − 2 2 2 2 2 2 (2.28) We can construct a larger group from two or more groups Gi , by means of certain products. A rather simple one is the direct product. We consider, e.g., two groups G1 and G2 . Their direct product is denoted G1 × G2 , and its multiplication rule is defined as (a1 , a2 )(b1 , b2 ) = (a1 b1 , a2 b2 ),

(2.29)

for a1 , b1 ∈ G1 and a2 , b2 ∈ G2 . The semi-direct product is a less trivial product between two groups G1 and G2 , and it is defined such that   (a1 , a2 )(b1 , b2 ) = a1 fa2 (b1 ), a2 b2 , (2.30) for a1 , b1 ∈ G1 and a2 , b2 ∈ G2 , where fa2 (b1 ) denotes a homomorphic map from G2 to the automorphisms of G1 . This semi-direct product is denoted by G1 f G2 . We consider the group G with a subgroup H and a normal subgroup N , whose elements are denoted hi and nj , respectively. When G = N H = H N and N ∩ H = {e}, the semi-direct product N f H is isomorphic to G, G  N f H , where we use the map f defined by fhi (nj ) = hi nj (hi )−1 .

(2.31)

For the notation of the semi-direct product, we will often omit f and denote it simply by N  H . Example Let us study the semi-direct product Z3  Z2 . Here we denote the Z3 and Z2 generators by c and h, i.e., c3 = e and h2 = e. In this case, (2.31) can be written hch−1 = cm ,

(2.32)

where m = 0, because all the elements of Z3 can be written cm (the case m = 0 being inconsistent). When m = 1, the above relation is trivial and leads simply to the direct product Z3 × Z2 . Thus, only the case with m = 2 is non-trivial, i.e., hch−1 = c2 .

(2.33)

20

2

Basics of Finite Groups

Indeed, this algebra is isomorphic to S3 , and h and c are identified with a and ab, respectively. Similarly, we can consider Zn  Zm . When we denote the Zn and Zm generators by a and b, respectively, they satisfy a n = bm = e,

bab−1 = a k ,

(2.34)

where k = 0, although the case with k = 1 leads to the direct product Zn × Zm .

References 1. Ramond, P.: Group Theory: A Physicist’s Survey. Cambridge University Press, Cambridge (2010) 2. Miller, G.A., Dickson, H.F., Blichfeldt, L.E.: Theory and Applications of Finite Groups. Wiley, New York (1916) 3. Hamermesh, M.: Group Theory and Its Application to Physical Problems. Addison-Wesley, Reading (1962) 4. Georgi, H.: Front. Phys. 54, 1 (1982) 5. Ludl, P.O.: arXiv:0907.5587 [hep-ph] 6. Grimus, W., Ludl, P.O.: J. Phys. A 45, 233001 (2012). arXiv:1110.6376 [hep-ph]

Chapter 3

SN

As introduced in the previous section, the symmetric group SN consists of all possible permutations among N objects xi with i = 1, . . . , N : (x1 , . . . , xN ) → (xi1 , . . . , xiN ).

(3.1)

The group S2 consists of two permutations (x1 , x2 ) → (x1 , x2 ),

(x1 , x2 ) → (x2 , x1 ).

(3.2)

This group is nothing but the Abelian group Z2 . Therefore, we study simple examples for N = 3 and 4, that is, S3 and S4 .

3.1 S3 We begin with S3 . Since some aspects of S3 have already been discussed in Chap. 2, we summarize them briefly and go on to study other aspects such as tensor products. This group consists of all permutations among three objects (x1 , x2 , x3 ), and so has order 3! = 6. All six elements can be written as products of elements a and b: a : (x1 , x2 , x3 ) → (x2 , x1 , x3 ), b : (x1 , x2 , x3 ) → (x3 , x2 , x1 ), together with the identity e, that is, {e, a, b, ab, ba, bab}.

(3.3)

3.1.1 Conjugacy Classes As studied in Chap. 2, S3 has the following three conjugacy classes: C1 : {e},

C2 : {ab, ba},

C3 : {a, b, bab}.

H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_3, © Springer-Verlag Berlin Heidelberg 2012

(3.4) 21

22

3

SN

Their orders are found from (ab)3 = (ba)3 = e,

a 2 = b2 = (bab)2 = e.

(3.5)

The elements {e, ab, ba} correspond to even permutations, and {a, b, bab} to odd permutations.

3.1.2 Characters and Representations The characters and representations are studied in Chap. 2. The group S3 has two singlet representations 1 and 1 , and a doublet 2. Their characters are summarized in Table 2.1. The characters for the singlets correspond to the representations on the singlets and the doublet representations are obtained in (2.28).

3.1.3 Tensor Products Here, we consider tensor products of irreducible representations. Let us start by discussing the tensor products of two doublets (x1 , x2 ) and (y1 , y2 ). For example, each element xi yj transforms under b according to √ x1 y1 + 3x2 y2 + 3(x1 y2 + x2 y1 ) , x1 y1 → 4 √ √ 3x1 y1 − 3x2 y2 − x1 y2 + 3x2 y1 x1 y2 → , 4 (3.6) √ √ 3x1 y1 − 3x2 y2 − x2 y1 + 3x1 y2 x2 y1 → , 4 √ 3x1 y1 + x2 y2 − 3(x1 y2 + x2 y1 ) x2 y2 → . 4 Thus, it is found that b(x1 y1 + x2 y2 ) = (x1 y1 + x2 y2 ),

b(x1 y2 − x2 y1 ) = −(x1 y2 − x2 y1 ). (3.7)

Therefore, these linear combinations correspond to the singlets 1 : x1 y2 − x2 y1 .

1 : x1 y1 + x2 y2 ,

(3.8)

Furthermore, it is found that 

x y − x1 y 1 b 2 2 x 1 y 2 + x2 y 1



 =

− 12 −

√

3 2

−

√

3 2

1 2



 x 2 y 2 − x1 y 1 . x 1 y 2 + x2 y 1

(3.9)

3.1 S3

23

Hence, (x2 y2 − x2 y2 , x1 y2 + x2 y1 ) corresponds to the doublet, i.e.,   x2 y2 − x1 y1 . 2= x1 y2 + x2 y1

(3.10)

Similarly, we can study the tensor product of the doublet (x1 , x2 ) and the singlet 1 : y  . Their products xi y  transform under b according to √ 3 1  x2 y  , x 1 y → x1 y + 2 2 √ 1 3  x1 y  − x2 y  . x2 y → 2 2 

(3.11)

They thus form a doublet  2:

 −x2 y  . x1 y 

(3.12)

These tensor products are summarized as follows:       x1 y1 x1 y2 + x2 y1  ⊗ = (x1 y1 + x2 y2 )1 + (x1 y2 − x2 y1 )1 + , x2 2 y2 2 x1 y1 − x2 y2 2         x1 x1 y  x1 −x2 y    ⊗ (y  )1 = , ⊗ (y ) = , 1 x2 2 x2 y  2 x2 2 x1 y  2 (x)1 ⊗ (y)1 = (xy)1 ,

(x)1 ⊗ (y)1 = (xy)1 .

(3.13) Obviously, the tensor product of two trivial singlets is a trivial singlet. Tensor products are important for applications to particle phenomenology. Matter and Higgs fields may be constructed to carry certain representations of discrete symmetries. The Lagrangian must be invariant under discrete symmetries. This implies that n-point couplings corresponding to a trivial singlet can appear in the Lagrangian. In addition to the above (real) representation of S3 , another representation, i.e., the complex representation, is often used in the literature. Let us consider changes of representation basis. The permutations in S3 in (3.1) are represented on the reducible triplet (x1 , x2 , x3 ) by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 1 0 0 0 1 0 ⎝0 1 0⎠, ⎝0 0 1⎠, ⎝1 0 0⎠, 0 0 1 0 1 0 0 0 1 (3.14) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 1 0 0 1 ⎝0 0 1⎠, ⎝0 1 0⎠, ⎝1 0 0⎠. 1 0 0 1 0 0 0 1 0

24

3

SN

We change the representation through the unitary transformation, U † gU , e.g., using the unitary matrix √ 1/√3 U = ⎝ 1/√3 1/ 3

√

⎛

2/3 √ −1/√6 −1/ 6

⎞ 0√ −1/√ 2 ⎠ . 1/ 2

(3.15)

Then, the six elements of S3 are written as ⎛

⎞ 1 0 0 ⎝0 1 0⎠, 0 0 1 ⎛ ⎞ 1 0 0√ ⎜0 −1 − 3 ⎟ ⎝ 2 2 ⎠, 0

√ 3 2

− 12

⎛ ⎞ 1 1 0 0 ⎜0 ⎝0 1 0 ⎠, ⎝ 0 0 −1 0 ⎛ ⎞ 1 0 0 √ ⎜0 −1 3⎟ ⎝ 2 2 ⎠, ⎛

√

0

3 2

⎞ 0√ − 23 ⎟ ⎠,

0 − 12 √

1 − 23 2 ⎛ 1 0 ⎜0 −1 ⎝ 2

1 2

0

−

√

3 2

0

√ 3 2 − 12

⎞

(3.16)

⎟ ⎠.

Note that this form is completely reducible and that the (bottom right) (2 × 2) submatrices are exactly the same as those for the doublet representation (2.28). We can use another unitary matrix U to obtain a completely reducible form from the reducible representation matrices (3.14). For example, we can use the unitary matrix ⎛ 1 1 1 ⎝ 1 w Uw = √ 3 1 w2

⎞ 1 w2 ⎠ , w

(3.17)

which is called the magic matrix. Then, the six elements of S3 are ⎛

⎛

1 ⎝0 0

1 0 ⎝0 w 0 0

0 1 0

⎞ 0 0⎠, 1 ⎞ 0 0 ⎠, w2

⎛

1 ⎝0 0 ⎛ 1 ⎝0 0

⎞ 0 1⎠, 0 ⎞ 0 0 0 w⎠, w2 0

0 0 1

⎛

⎞ 1 0 0 ⎝ 0 0 w2 ⎠ , 0 w 0 ⎛ ⎞ 1 0 0 ⎝ 0 w2 0 ⎠ . 0 0 w

(3.18)

The (bottom right) (2 × 2) submatrices correspond to the doublet representation in the different basis, that is, the complex representation. In different bases, the multiplication rule does not change. For example, we obtain 2 × 2 = 1 + 1 + 2 in both the real and complex bases. However, elements of doublets on the left-hand side are written in a different way.

3.2 S4

25

Fig. 3.1 The S4 symmetry of a cube. The figure shows the transformations corresponding to the S4 elements with h = 2, 3, and 4. Note that the group can also be considered as the symmetries of a regular octahedron in a way similar to the cube

3.2 S4 We now discuss the group S4 , which consists of all permutations among four objects (x1 , x2 , x3 , x4 ): (x1 , x2 , x3 , x4 )

→

(xi , xj , xk , xl ).

(3.19)

The order of S4 is 4! = 24. We denote all the elements of S4 as follows: a1 : (x1 , x2 , x3 , x4 ),

a2 : (x2 , x1 , x4 , x3 ),

a3 : (x3 , x4 , x1 , x2 ),

a4 : (x4 , x3 , x2 , x1 ),

b1 : (x1 , x4 , x2 , x3 ),

b2 : (x4 , x1 , x3 , x2 ),

b3 : (x2 , x3 , x1 , x4 ),

b4 : (x3 , x2 , x4 , x1 ),

c1 : (x1 , x3 , x4 , x2 ),

c2 : (x3 , x1 , x2 , x4 ),

c3 : (x4 , x2 , x1 , x3 ),

c4 : (x2 , x4 , x3 , x1 ),

d1 : (x1 , x2 , x4 , x3 ),

d2 : (x2 , x1 , x3 , x4 ),

d3 : (x4 , x3 , x1 , x2 ),

d4 : (x3 , x4 , x2 , x1 ),

e1 : (x1 , x3 , x2 , x4 ),

e2 : (x3 , x1 , x4 , x2 ),

e3 : (x2 , x4 , x1 , x3 ),

e4 : (x4 , x2 , x3 , x1 ),

f1 : (x1 , x4 , x3 , x2 ),

f2 : (x4 , x1 , x2 , x3 ),

f3 : (x3 , x2 , x1 , x4 ),

f4 : (x2 , x3 , x4 , x1 ),

(3.20)

where we have shown the ordering of four objects after permutations. S4 describes the symmetries of a cube, as shown in Fig. 3.1. It is obvious that x1 + x2 + x3 + x4 is invariant under any permutation of S4 , so it is a trivial singlet. Thus, we use the vector space which is orthogonal to this singlet direction, viz., ⎛ ⎞ ⎛ ⎞ Ax x1 + x2 − x3 − x4 3 : ⎝ Ay ⎠ = ⎝ x1 − x2 + x3 − x4 ⎠ , (3.21) Az x1 − x2 − x3 + x4

26

3

SN

to construct matrix representations of S4 , that is, a triplet representation. In this triplet vector space, the elements of S4 are represented by the following matrices: ⎛

⎞ 1 0 0 a1 = ⎝ 0 1 0 ⎠ , 0 0 1 ⎛

−1 0 a3 = ⎝ 0 1 0 0

⎞ 0 0 ⎠, −1

⎛

⎞ 0 0 1 b1 = ⎝ 1 0 0 ⎠ , 0 1 0 ⎛

⎞ 0 0 −1 0 ⎠, b3 = ⎝ 1 0 0 −1 0 ⎛

⎞ 0 1 0 c1 = ⎝ 0 0 1 ⎠ , 1 0 0 ⎛

⎞ 0 −1 0 0 1⎠, c3 = ⎝ 0 −1 0 0 ⎛

⎞ 1 0 0 d1 = ⎝ 0 0 1 ⎠ , 0 1 0 ⎛

⎞ −1 0 0 0 1⎠, d3 = ⎝ 0 0 −1 0 ⎛

⎞ 0 1 0 e1 = ⎝ 1 0 0 ⎠ , 0 0 1 ⎛

⎞ 0 −1 0 0 ⎠, e3 = ⎝ 1 0 0 0 −1

⎛

⎞ 1 0 0 a2 = ⎝ 0 −1 0 ⎠ , 0 0 −1 ⎛

⎞ −1 0 0 a4 = ⎝ 0 −1 0 ⎠ , 0 0 1 ⎛

⎞ 0 0 1 b2 = ⎝ −1 0 0 ⎠ , 0 −1 0 ⎛

0 0 b4 = ⎝ −1 0 0 1

⎞ −1 0 ⎠, 0

⎛

⎞ 0 1 0 c2 = ⎝ 0 0 −1 ⎠ , −1 0 0 ⎛

0 c4 = ⎝ 0 1

⎞ −1 0 0 −1 ⎠ , 0 0

⎛

⎞ 1 0 0 d2 = ⎝ 0 0 −1 ⎠ , 0 −1 0 ⎛

−1 0 d4 = ⎝ 0 0 0 1 ⎛

0 1 e2 ⎝ −1 0 0 0 ⎛

⎞ 0 −1 ⎠ , 0

⎞ 0 0 ⎠, −1

⎞ 0 −1 0 e4 = ⎝ −1 0 0 ⎠ , 0 0 1

(3.22)

3.2

27

S4

⎛

0 f1 = ⎝ 0 1

0 1 0

⎞ 1 0⎠, 0

⎛

0 f2 = ⎝ 0 −1

⎞ −1 0 ⎠, 0

0 0 f3 = ⎝ 0 1 −1 0

⎛

⎛

0 f4 = ⎝ 0 1

⎞ 0 1 −1 0 ⎠ , 0 0 ⎞ 0 −1 −1 0 ⎠ . 0 0

3.2.1 Conjugacy Classes The elements of S4 can be classified by their order h, i.e., the smallest positive integer such that a h = e: h = 1: {a1 }, h = 2: {a2 , a3 , a4 , d1 , d2 , e1 , e4 , f1 , f3 }, h = 3: {b1 , b2 , b3 , b4 , c1 , c2 , c3 , c4 },

(3.23)

h = 4: {d3 , d4 , e2 , e3 , f2 , f4 }. Moreover, they are classified by the conjugacy classes according to: C1 :

{a1 },

h = 1,

C3 :

{a2 , a3 , a4 },

h = 2,

C6 :

{d1 , d2 , e1 , e4 , f1 , f3 },

h = 2,

C8 :

{b1 , b2 , b3 , b4 , c1 , c2 , c3 , c4 }, h = 3,

C6 :

{d3 , d4 , e2 , e3 , f2 , f4 },

(3.24)

h = 4.

3.2.2 Characters and Representations The group S4 includes five conjugacy classes, so there are five irreducible representations. For example, all its elements can be written as products of b1 in C8 and d4 in C6 , which satisfy (b1 )3 = e,

(d4 )4 = e,

d4 (b1 )2 d4 = b1 ,

d4 b1 d4 = b1 (d4 )2 b1 . The orthogonality relation (2.18) requires 2 χα (C1 ) = mn n2 = m1 + 4m2 + 9m3 + · · · = 24, α

n

(3.25)

(3.26)

28

3

Table 3.1 Characters of S4 representations

h

χ1

χ 1

χ2

χ3

SN χ 3

C1

1

1

1

2

3

3

C3

2

1

1

2

−1

−1

C6

2

1

−1

0

1

−1

C6

4

1

−1

0

−1

1

C8

3

1

1

−1

0

0

like (2.20), and mn also satisfy m1 + m2 + m3 + · · · = 5, because there must be five irreducible representations. Then, we can easily find the unique solution as (m1 , m2 , m3 ) = (2, 1, 2). Therefore, the irreducible representations of S4 include two singlets 1 and 1 , one doublet 2, and two triplets 3 and 3 , where 1 corresponds to a trivial singlet and 3 corresponds to (3.21) and (3.22). We can compute the character for each representation by a similar analysis to the one adopted for S3 . The characters are shown in Table 3.1. For 2, the representation matrices are, for example,  a2 (2) =

 1 0 , 0 1

 b1 (2) = 

d1 (2) = d3 (2) = d4 (2) =

0 1

ω 0

 1 . 0

 0 , ω2 (3.27)

For 3 , the representation matrices are, for example, ⎛

⎞ 1 0 0 a2 (3 ) = ⎝ 0 −1 0 ⎠ , 0 0 −1 ⎛ ⎞ −1 0 0 0 −1 ⎠ , d1 (3 ) = ⎝ 0 0 −1 0 ⎛ ⎞ 1 0 0 d4 (3 ) = ⎝ 0 0 1 ⎠ . 0 −1 0

⎛

⎞ 0 0 1 b1 (3 ) = ⎝ 1 0 0 ⎠ , 0 1 0 ⎛ ⎞ 1 0 0 d3 (3 ) = ⎝ 0 0 −1 ⎠ , 0 1 0

(3.28)

Note that a2 (3 ) = a2 (3) and b1 (3 ) = b1 (3), but d1 (3 ) = −d1 (3), d3 (3 ) = −d3 (3), and d4 (3 ) = −d4 (3). This aspect should be obvious from the above character table.

3.2

29

S4

3.2.3 Tensor Products Finally, we present the tensor products. The tensor products of 3 × 3 can be decomposed as   A·Σ ·B (A)3 × (B)3 = (A · B)1 + A · Σ∗ · B 2 ⎞ ⎞ ⎛ ⎛ {Ay Bz } [Ay Bz ] + ⎝ {Az Bx } ⎠ + ⎝ [Az Bx ] ⎠ , (3.29) {Ax By } 3 [Ax By ] 3 where A · B = Ax Bx + Ay By + Az Bz , {Ai Bj } = Ai Bj + Aj Bi , [Ai Bj ] = Ai Bj − Aj Bi ,

(3.30)

A · Σ · B = Ax Bx + ωAy By + ω2 Az Bz , A · Σ ∗ · B = Ax Bx + ω2 Ay By + ωAz Bz . The tensor products of other representations can be decomposed in a similar way, e.g., ⎛ ⎛ ⎞ ⎞   {Ay Bz } [Ay Bz ] A·Σ ·B + ⎝ {Az Bx } ⎠ + ⎝ [Az Bx ] ⎠ , (A)3 × (B)3 = (A · B)1 + A · Σ∗ · B 2 {Ax By } 3 [Ax By ] 3 (3.31) 

A·Σ ·B (A)3 × (B)3 = (A · B)1 + −A · Σ ∗ · B

⎞ ⎞ ⎛ {Ay Bz } [Ay Bz ] + ⎝ {Az Bx } ⎠ + ⎝ [Az Bx ] ⎠ , 2 {Ax By } 3 [Ax By ] 3 (3.32)



⎛

and

 (A)2 × (B)2 = {Ax By }1 + [Ax By ]1 + 

Ax Ay



⎛

Ay By Ax Bx

 ,

(3.33)

2

⎛ ⎛ ⎞ ⎞ ⎞ (Ax + Ay )Bx (Ax − Ay )Bx Bx × ⎝ By ⎠ = ⎝ (ω2 Ax + ωAy )By ⎠ + ⎝ (ω2 Ax − ωAy )By ⎠ , 2 Bz 3 (ωAx + ω2 Ay )Bz 3 (ωAx − ω2 Ay )Bz 3 (3.34)

30

3



Ax Ay



SN

⎛

⎛ ⎛ ⎞ ⎞ ⎞ (Ax + Ay )Bx (Ax − Ay )Bx Bx × ⎝ By ⎠ = ⎝ (ω2 Ax + ωAy )By ⎠ + ⎝ (ω2 Ax − ωAy )By ⎠ . 2 Bz 3 (ωAx + ω2 Ay )Bz 3 (ωAx − ω2 Ay )Bz 3 (3.35)

In addition, we have decompositions 3 × 1 = 3 , 3 × 1 = 3, and 2 × 1 = 2. In the literature,several bases are used for S4 . The decomposition of tensor products, r × r = m rm , does not depend on the basis. For example, we obtain 3 × 3 = 1 + 2 + 3 + 3 in any basis. However, the multiplication rules written in terms of components do depend on the basis we use. Here we have used the basis (3.27). In Appendix B, we show the relations between several bases and give the multiplication rules explicitly in terms of components. Similarly, we can study the group SN with N > 4. Here we give a brief comment on such groups. The group SN with N > 4 has only one invariant subgroup, that is, the alternating group AN . SN has two one-dimensional representations: one is the trivial singlet, that is, invariant under all elements (symmetric representation), the other is a pseudo-singlet, that is, symmetric under the even permutation elements, but antisymmetric under the odd permutation elements. Group-theoretical aspects of S5 are derived from those of S4 by applying a theorem due to Frobenius (Frobenius formula), a graphical method (Young tableaux), and recursion formulas for characters (branching laws). The details can be found in the textbook [1], for example. Such analysis can be extended recursively from SN to SN +1 .

References 1. Hamermesh, M.: Group Theory and Its Application to Physical Problems. Addison-Wesley, Reading (1962)

Chapter 4

AN

In this chapter, we study the group AN , consisting of all even permutations in SN . These do indeed form a group, also called the alternating group. The order of this group is clearly (N!)/2. Let us consider a simple example. As discussed in Sect. 3.1, the even permutations in S3 are e : (x1 , x2 , x3 ) → (x1 , x2 , x3 ), a4 : (x1 , x2 , x3 ) → (x3 , x1 , x2 ),

(4.1)

a5 : (x1 , x2 , x3 ) → (x2 , x3 , x1 ), while the odd permutations are a1 : (x1 , x2 , x3 ) → (x2 , x1 , x3 ), a2 : (x1 , x2 , x3 ) → (x3 , x2 , x1 ),

(4.2)

a3 : (x1 , x2 , x3 ) → (x1 , x3 , x2 ). The three even permutations {e, a4 , a5 } form the group A3 . Since (a4 )2 = a5 and (a4 )3 = e, the group A3 is nothing but Z3 . Therefore, we start by studying A4 , the smallest non-Abelian group.

4.1 A4 The group A4 consists of all even permutations in S4 and thus has order (4!)/2 = 12. A4 is the symmetry group of a tetrahedron as shown in Fig. 4.1. Indeed, the group A4 is often denoted by T . Using the notation in Sect. 3.2, the 12 elements are: H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_4, © Springer-Verlag Berlin Heidelberg 2012

31

32

4

AN

Fig. 4.1 The A4 symmetry of the tetrahedron

⎛

⎛ ⎞ ⎞ 1 0 0 1 0 0 a1 = ⎝ 0 1 0 ⎠ , a2 = ⎝ 0 −1 0 ⎠ , 0 0 1 0 0 −1 ⎛ ⎛ ⎞ ⎞ −1 0 0 −1 0 0 a3 = ⎝ 0 1 0 ⎠ , a4 = ⎝ 0 −1 0 ⎠ , 0 0 −1 0 0 1 ⎛ ⎛ ⎞ ⎞ 0 0 1 0 0 1 b1 = ⎝ 1 0 0 ⎠ , b2 = ⎝ −1 0 0 ⎠ , 0 1 0 0 −1 0 ⎛ ⎛ ⎞ ⎞ 0 0 −1 0 0 −1 0 ⎠, b3 = ⎝ 1 0 b4 = ⎝ −1 0 0 ⎠ , 0 −1 0 0 1 0 ⎛ ⎛ ⎞ ⎞ 0 1 0 0 1 0 c1 = ⎝ 0 0 1 ⎠ , c2 = ⎝ 0 0 −1 ⎠ , 1 0 0 −1 0 0 ⎛ ⎛ ⎞ ⎞ 0 −1 0 0 −1 0 0 1⎠, c4 = ⎝ 0 0 −1 ⎠ . c3 = ⎝ 0 −1 0 0 1 0 0

(4.3)

As can be seen from this, A4 is obviously isomorphic to Δ(12)  (Z2 × Z2 )  Z3 (see Chap. 10 for explanation). They are classified into conjugacy classes as follows: C1 : C3 : C4 : C4 :

{a1 }, {a2 , a3 , a4 }, {b1 , b2 , b3 , b4 }, {c1 , c2 , c3 , c4 },

h = 1, h = 2, h = 3, h = 3,

(4.4)

where we have also shown the order h of each element in the conjugacy class. There are four conjugacy classes and there must therefore be four irreducible representations, i.e., m1 + m2 + m3 + · · · = 4.

4.1

33

A4

The orthogonality relation (2.17) requires  2  χα (C1 ) = mn n2 = m1 + 4m2 + 9m3 + · · · = 12, α

(4.5)

n

for the mi , which must also satisfy m1 + m2 + m3 + · · · = 4. The only solution is (m1 , m2 , m3 ) = (3, 0, 1). That is, the A4 group has three singlets, 1, 1 , and 1 , and a single triplet 3, where the triplet corresponds to (4.3). Another algebraic definition of A4 is often used in the literature. Let a1 = e, a2 = s, and b1 = t. They satisfy the algebraic relations s 2 = t 3 = (st)3 = e.

(4.6)

The closed algebra of these elements s and t is defined as A4 . It is straightforward to write all the ai , bi , and ci in terms of s and t. Then, the conjugacy classes can be reexpressed as C1 :

{e},

h = 1,

C3 :

{s, tst 2 , t 2 st},

C4 :

{t, ts, st, sts},

h = 2, h = 3,

C4 :

{t 2 , st 2 , t 2 s, tst},

h = 3.

(4.7)

We now use these to study the characters. First, we consider the characters of the three singlets. Because s 2 = e, the characters of C3 have two possibilities, namely χα (C3 ) = ±1. However, the two elements t and ts belong to the same conjugacy class C4 . This means that χα (C3 ) should have the value χα (C3 ) = 1. Similarly, because t 3 = e, the characters χα (t) can correspond to three possible values, i.e., χα (t) = ωn , n = 0, 1, 2, and all three values are consistent with the above conjugacy class structure. Thus, the three singlets 1, 1 , and 1 are classified by the three values χα (t) = 1, ω, and ω2 , respectively. Clearly, χα (C4 ) = [χα (C4 )]2 . Thus, generators such as s = a2 , t = b1 , and t 2 = c1 are represented on the non-trivial singlets 1 and 1 by s(1 ) = a2 (1 ) = 1, s(1 ) = a2 (1 ) = 1,

t (1 ) = b1 (1 ) = ω, t (1 ) = b1 (1 ) = ω2 ,

t 2 (1 ) = c1 (1 ) = ω2 , t 2 (1 ) = c1 (1 ) = ω.

(4.8)

These characters are shown in Table 4.1. Next, we consider the characters for the triplet representation. Obviously, the matrices in (4.3) correspond to the triplet representation. We thus obtain their characters and the results are also shown in Table 4.1. The tensor product of 3 × 3 can be decomposed as

(A)3 × (B)3 = (A · B)1 + (A · Σ · B)1 + A · Σ ∗ · B 1 ⎛ ⎛ ⎞ ⎞ {Ay Bz } [Ay Bz ] + ⎝ {Az Bx } ⎠ + ⎝ [Az Bx ] ⎠ . (4.9) {Ax By } 3 [Ax By ] 3

34

4

Table 4.1 Characters of A4 representations

h

χ1

χ1 

χ1

AN χ3

C1

1

1

1

1

3

C3

2

1

1

1

−1

C4

3

1

ω

ω2

0

1

ω2

ω

0

C4

3

Fig. 4.2 The regular icosahedron

4.2 A5 Next we study the group A5 . This group is isomorphic to the symmetry group of a regular icosahedron. Thus, it is of pedagogical interest to explain the grouptheoretical aspects of A5 in terms of the symmetries of a regular icosahedron [1]. As shown in Fig. 4.2, a regular icosahedron consists of 20 identical equilateral triangular faces, 30 edges and 12 vertices. The icosahedron is dual to a dodecahedron, whose symmetry group is also isomorphic to A5 . The elements of A5 correspond to all the proper rotations of the icosahedron. Such rotations are classified into five types, that is, the 0 rotation (identity), rotations by π about the midpoint of each edge, rotations by 2π/3 about axes through the center of each face, and rotations by 2π/5 and 4π/5 about an axis through each vertex. Following [1], we label the vertices by n = 1, . . . , 12 in Fig. 4.2. Here, we define two elements a and b such that a corresponds to the rotation by π about the midpoint of the edge between vertices 1 and 2 while b corresponds to the rotation by 2π/3 about the axis through the center of the triangular face 10-11-12. That is, these two elements correspond to the transformations acting on the 12 vertices as follows: a : (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) → (2, 1, 4, 3, 8, 9, 12, 5, 6, 11, 10, 7), b : (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) → (2, 3, 1, 5, 6, 4, 8, 9, 7, 11, 12, 10). Then the product ab is given by the transformation ab : (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) → (3, 2, 5, 1, 9, 7, 10, 6, 4, 12, 11, 8),

4.2

35

A5

which is the rotation by 2π/5 about the axis through vertex 2. All the elements of A5 can be written in terms of products of these elements, which satisfy a 2 = b3 = (ab)5 = e.

(4.10)

4.2.1 Conjugacy Classes The order of A5 is (5!)/2 = 60. The elements of A5 , i.e., all the rotations of the icosahedron, are classified into five conjugacy classes as follows: C1 : C15 : C20 : C12 :  : C12

{e},  a(12), a(13), a(14), a(16), a(18), a(23), a(24), a(25), a(29), a(35), a(36), a(37), a(48), a(49), a(59) ,  b(123), b(124), b(126), b(136), b(168), b(235), b(249), b(259), b(357), b(367), and their inverse elements ,  c(1), c(2), c(3), c(4), c(5), c(6), and their inverse elements ,  2 c (1), c2 (2), c2 (3), c2 (4), c2 (5), c2 (6), and their inverse elements , (4.11)

where a(km), b(kmn) and c(k) denote respectively the rotation by π about the midpoint of the edge km, the rotation by 2π/3 about the axis through the center of the face kmn, and the rotation by 2π/5 about the axis through the vertex k. The  contain 1, 15, 20, 12, and 12 elements, conjugacy classes C1 , C15 , C20 , C12 , and C12 2 respectively. Since obviously [a(km)] = [b(kmn)]3 = [c(k)]5 = e, we find h = 2  , where h denotes the order of in C15 , h = 3 in C20 , h = 5 in C12 , and h = 5 in C12 h each element in the conjugacy class, i.e., g = e.

4.2.2 Characters and Representations The orthogonality relations (2.18) and (2.19) for A5 are m1 + 4m2 + 9m3 + 16m4 + 25m5 + · · · = 60, m1 + m2 + m3 + m4 + m5 + · · · = 5.

(4.12) (4.13)

Solving these equations, we obtain (m1 , m2 , m3 , m4 , m5 ) = (1, 0, 2, 1, 1). Therefore, the A5 group has one trivial singlet 1, two triplets 3 and 3 , one quartet 4, and one quintet 5. The characters are shown in Table 4.2. There are several ways to construct these representations, e.g., the Cummins– Patera basis [2], the Shirai basis [1], or the Feruglio–Paris basis [3]. Each of the relations is also summarized in [3]. Instead of a and b, which is called the Cummins– Patera basis [2], we use the generators of Shirai’s basis [1], viz., s = a and t = bab, which satisfy

3 s 2 = t 5 = t 2 st 3 st −1 stst −1 = e. (4.14)

36

4

Table 4.2 Characters of A5 representations, where √ φ = (1 + 5)/2

AN

h

χ1

χ3

χ3 

χ4

χ5

C1

1

1

3

3

4

5

C15

2

1

−1

−1

0

1

C20

3

1

0

0

1

−1

C12

5

1

φ

1−φ

−1

0

 C12

5

1

1−φ

φ

−1

0

The generators s and t are represented by [1] ⎛ ⎛ ⎞ ⎞ 1 −1 φ 1 φ φ1 φ 1⎜ 1⎜ ⎟ ⎟ 1 ⎠, 1 ⎠ , on 3, s = ⎝ φ φ1 t = ⎝ −φ φ1 2 2 1 1 1 −φ −1 φ φ φ ⎛ ⎛ ⎞ ⎞ 1 1 −φ φ1 −φ − φ1 1⎜ 1⎜ ⎟ ⎟ −1 φ ⎠ , 1 φ ⎠ , on 3 , s = ⎝ φ1 t = ⎝ φ1 2 2 1 φ φ1 −1 φ − φ1 √ ⎞ ⎛ −1 −1 −3 −√5 1 ⎜ −1 3 1 −√ 5 ⎟ ⎟, s= ⎜ ⎝ ⎠ −3 1 −1 5 4 √ √ √ − 5 − 5 5 −1 √ ⎞ ⎛ −1 1 −3 √5 ⎟ 1 ⎜ −1 −3 1 √5 ⎟ , on 4, t= ⎜ ⎝ 5⎠ 1 1 4 √3 √ √ 5 − 5 − 5 −1 ⎞ ⎛ √ √ φ2 1−3φ 5 3 1 − 2 2 4φ ⎟ 2φ 2 ⎜ 42 √ ⎜ φ 3 ⎟ 1 1 0 ⎟ ⎜ 2 2φ √ ⎟ 1⎜ 3φ ⎟ , 1 s= ⎜ − 2 1 0 −1 − 2 ⎟ 2⎜ √ ⎟ ⎜ √2φ 3 ⎟ ⎜ 5 0 −1 1 − ⎝ √2 2 ⎠ √ √ √ 3φ 3φ−1 3 3 − 2 − 23 4φ 2φ 4 ⎛ √ ⎞ √ 2 1−3φ φ 5 3 1 − − − 2 2 4φ ⎟ 2φ 2 ⎜ 42 √ ⎜ φ 3 ⎟ −1 1 0 ⎜ 2 2φ ⎟ √ ⎟ ⎜ 1 3φ ⎟ , on 5, 1 t= ⎜ 1 0 −1 ⎟ ⎜ 2 2 ⎜ 2φ√ √2 ⎟ 3 ⎟ ⎜− 5 0√ 1 1 ⎝ √2 2 ⎠ √ √ 3φ−1 3φ 3 3 3 − − 4φ 2φ 2 2 4 √ where φ = (1 + 5)/2. The multiplication rules are shown in Table 4.3.

(4.15)

(4.16)

(4.17)

(4.18)

4.2 A5 Table 4.3 Multiplication rules for the group A5

37 3⊗3=1⊕3⊕5 3 ⊗ 3 = 1 ⊕ 3 ⊕ 5 3 ⊗ 3 = 4 ⊕ 5 3 ⊗ 4 = 3 ⊕ 4 ⊕ 5 3 ⊗ 4 = 3 ⊕ 4 ⊕ 5 3 ⊗ 5 = 3 ⊕ 3 ⊕ 4 ⊕ 5 3 ⊗ 5 = 3 ⊕ 3 ⊕ 4 ⊕ 5 4 ⊗ 4 = 1 ⊕ 3 ⊕ 3 ⊕ 4 ⊕ 5 4 ⊗ 5 = 3 ⊕ 3 ⊕ 4 ⊕ 5 ⊕ 5 5 ⊗ 5 = 1 ⊕ 3 ⊕ 3 ⊕ 4 ⊕ 4 ⊕ 5 ⊕ 5

4.2.3 Tensor Products Concrete tensor products are partially available in [4], using the Shirai basis. Here we display the full set of tensor products: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x1 y1 x3 y2 − x2 y3 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = (x1 y1 + x2 y2 + x3 y3 )1 ⊕ ⎝ x1 y3 − x3 y1 ⎠ x3 3 y3 3 x2 y1 − x1 y2 3 ⎞ ⎛ x2 y2 − x1 y1 ⎟ ⎜ x2 y1 + x1 y2 ⎟ ⎜ ⎟ , x3 y2 + x2 y3 ⊕⎜ (4.19) ⎟ ⎜ ⎠ ⎝ x1 y3 + x3 y1 − √1 (x1 y1 + x2 y2 − 2x3 y3 ) 3 5 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ x1 y1 x3 y2 − x2 y3 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = (x1 y1 + x2 y2 + x3 y3 )1 ⊕ ⎝ x1 y3 − x3 y1 ⎠ x3 3 y3 3 x2 y1 − x1 y2 3 ⎞ ⎛ √ 1 1 2 (− φ x1 y1 − φx2 y2 + 5x3 y3 ) ⎟ ⎜ x2 y1 + x1 y2 ⎟ ⎜ ⎟ ⎜ −(x3 y1 + x1 y3 ) ⊕⎜ ⎟ , ⎟ ⎜ x2 y3 + x3 y2 ⎠ ⎝ 1 √ [(1 − 3φ)x1 y1 + (3φ − 2)x2 y2 + x3 y3 ] 2 3 5 ⎛ ⎞ ⎛ ⎞ y1 x1 ⎝ x2 ⎠ ⊗ ⎝ y 2 ⎠ x3 3 y3 3 √ ⎛1 2 ⎞ 1 ⎛ 1 ⎞ 2 (φ x2 y1 + φ 2 x1 y2 − 5x3 y3 ) ⎜ ⎟ φ x3 y2 − φx1 y3 −(φx1 y1 + φ1 x2 y2 ) ⎜ ⎟ ⎜ ⎟ 1 ⎜ ⎟ ⎜ φx3 y1 + φ x2 y3 ⎟ 1 ⎜ ⎟ , x y − φx y =⎜ ⊕ (4.20) ⎟ 2 3 φ 3 1 ⎜ ⎟ 1 ⎝ − φ x1 y1 + φx2 y2 ⎠ 1 ⎜ ⎟ φx3 y2 + φ x1 y3 ⎝ √ ⎠ x2 y1 − x1 y2 + x3 y3 4 3 1 2 ( φ x2 y1 + φx1 y2 + x3 y3 ) 5

38

4

⎛ ⎞ ⎞ y1 x1 ⎜ y2 ⎟ ⎟ ⎜ ⎝ x2 ⎠ ⊗ ⎝ y3 ⎠ x3 3 y4 4 ⎞ ⎛ ⎞ ⎛ 1 −x1 y3 + x2 y4 − x3 y2 − φ 2 x1 y3 + φ1 x2 y4 + x3 y2 ⎜ −x1 y4 − x2 y3 + x3 y1 ⎟ ⎟ ⎜ ⎟ = ⎝ − φ1 x1 y4 + x2 y3 + φ12 x3 y1 ⎠ ⊕ ⎜ ⎝ x1 y1 + x2 y2 + x3 y4 ⎠ 1 1 −x1 y1 + φ 2 x2 y2 + φ x3 y4  x1 y2 − x2 y1 − x3 y3 4 3 ⎞ ⎛1 2 [(6φ + 5)x1 y2 + (3φ + 4)x2 y1 + (3φ + 1)x3 y3 ] ⎟ ⎜ −x1 y1 + (3φ + 2)x2 y2 − (3φ + 1)x3 y4 ⎟ ⎜ ⎟ , −(3φ + 1)x y − x y − (3φ + 2)x y ⊕⎜ 1 4 2 3 3 1 ⎟ ⎜ ⎠ ⎝ −(3φ + 2)x1 y3 − (3φ + 1)x2 y4 + x3 y2

AN

⎛

√

⎛

⎞

⎛

3 2 [x1 y2

⎞

− (3φ + 2)x2 y1 + 3(φ + 1)x3 y3 ]

(4.21)

5

y1 x1 ⎜ y2 ⎟ ⎝ x2 ⎠ ⊗ ⎜ ⎟ ⎝ y3 ⎠ x3 3 y4 4 ⎞ ⎛ ⎞ ⎛ x1 y4 − x2 y3 + x3 y2 x1 y3 + φx2 y4 + φ 2 x3 y1 ⎜ x1 y3 + x2 y4 − x3 y1 ⎟ ⎟ = ⎝ −φx1 y4 − φ 2 x2 y3 − x3 y2 ⎠ ⊕ ⎜ ⎝ −x1 y2 + x2 y1 + x3 y4 ⎠ 2 −φ x1 y2 − x2 y1 − φx3 y4 3 −(x1 y1 + x2 y2 + x3 y3 ) 4 ⎞ ⎛ 4 2 x1 y1 − φ x2 y2 + φ (2φ − 1)x3 y3 ⎜ x1 y2 − φ 4 x2 y1 + φ 2 (2φ − 1)x3 y4 ⎟ ⎟ ⎜ 4 2 ⎟ ⊕⎜ ⎜ φ x1 y3 − φ (2φ − 1)x2 y4 + x3 y1 ⎟ , ⎝ φ 2 (2φ − 1)x1 y4 − x2 y3 − φ 4 x3 y2 ⎠ √ − 3φ(φ 2 x1 y1 − x2 y2 − φx3 y3 ) 5 ⎛ ⎞ y1 ⎞ ⎛ ⎛ ⎞ x1 (y1 + √1 y5 ) − x2 y2 − x3 y4 ⎜ y2 ⎟ x1 3 ⎜ ⎟ ⎟ 1 ⎝ x2 ⎠ ⊗ ⎜ y 3 ⎟ = ⎜ ⎝ −x1 y2 − x2 (y1 − √3 y5 ) − x3 y3 ⎠ ⎜ ⎟ x3 3 ⎝ y 4 ⎠ −x1 y4 − x2 y3 − √2 x3 y5 3 3 y5 5 √ ⎞ ⎛ x1 y2 − φ2 x2 y1 − 2φ32 x2 y5 − φ12 x3 y3 ⎟ ⎜ √3 1 1 ⎟ ⊕⎜ ⎝ − 2 x1 y5 − 2φ 3 x1 y1√+ φ 2 x2 y2√− x3 y4 ⎠ − φ12 x1 y4 + x2 y3 + 2φ5 x3 y1 − 2φ3 x3 y5 

(4.22)

3

⎛ ⎜ ⎜ ⊕⎜ ⎜ ⎝

φ 2 −6

√

⎞

1 x y + 2 x2 y1 + 23 φ 2 x2 y5 + φ 2 x3 y3 φ2 1 2 √ ⎟ 3 − φ+4 x y − x y − φ 2 x2 y2 − φ12 x3 y4 ⎟ 1 1 2 ⎟ 2φ 2 1 5 √ √ ⎟ φ 2 x1 y4 + φ12 x2 y3 − 25 x3 y1 − 3 2 3 x3 y5 ⎠

√

5(x1 y3 + x2 y4 + x3 y2 )

4

4.2 A5

39

⎛

⎞ x1 y3 + x2 y4 − 2x3 y2 ⎜ ⎟ x1 y4 − x2 y3 + 2x3 y√1 ⎜ ⎟ ⎜ ⊕ ⎜ −x1 y1 + x2 y2 − x3 y4 + √3x1 y5 ⎟ ⎟ , ⎝ −x1 y2 − x2 y1 + x3 y3 − 3x2 y5 ⎠ √ − 3(x1 y3 − x2 y4 ) 5

(4.23)

⎞ √ y1 ⎞ ⎛ 1 ⎜ y2 ⎟ x2 y1 + 23 φ 2 x2 y5 + x3 y4 −φ 2 x1 y2 + 2φ x1 ⎜ ⎟ √ ⎟ 3 ⎝ x2 ⎠ ⊗ ⎜ y 3 ⎟ = ⎜ 2 ⎠ ⎝ 2φ+1 ⎜ ⎟ 2 x1 y1 + 2 x1√y5 − x2 y2 −√φ x3 y3 ⎠ ⎝ x3 3 y4 5 3 2 x1 y3 + φ x2 y4 − 2 φx3 y1 + 2 φx3 y5 3 y5 5 ⎛

⎞

⎛

⎛

⎞

3φ−1 √ x y 2 3 1 5 ⎟ ⎜ φ 3φ−2 −x y + x y − x y − √ x2 y5 ⎟ ⊕⎜ 2 3 ⎠ ⎝ 1 2 2 2 1√ 3 4 1 x1 y3 − x2 y4 − 25 x3 y1 − √ x3 y5 2 3 3

⎛ ⎜ ⎜ ⊕⎜ ⎜ ⎝

1 2φ x1 y1

− x2 y2 + x3 y3 +

⎞

√

√1 ( 12 x1 y1 + φ 2 x2 y2 + 12 x3 y3 − 3φx1 y5 ) φ 5 φ √ ⎟ √1 (− 12 x1 y2 − φ 2 x2 y1 − 3(φ−3) √ x2 y5 − φ 2 x3 y4 ) ⎟ ⎟ φ 5 5 √ √ ⎟ √1 (φ 2 x1 y3 − 12 x2 y4 + 5x3 y1 + 3x3 y5 ) ⎠ φ 5

x1 y4 − x2 y3 + x3 y2

⎞ ⎛ ⎞ y1 x1 ⎜ y2 ⎟ ⎜ x2 ⎟ ⎜ ⎟ ⊗⎜ ⎟ ⎝ y3 ⎠ ⎝ x3 ⎠ x4 4 y4 4 ⎛

4

⎞

⎛

−(3φ − 1)x1 y4 + (2 − 3φ)x2√y3 + x3 y2 ⎟ ⎜ −2x1 y3 − 2x2 y4 − x3 y1 + 15x √ 3 y5 ⎟ ⎜ ⎜ 2x1 y2 − (2 − 3φ)x2 y1 − 2x3 y4 + 3φx2 y5 ⎟ ⊕⎜ ⎟ , (4.24) √ ⎜ (3φ − 1)x y + 2x y + 2x y − 3 x y ⎟ 1 1 2 2 3 3 ⎝ φ 1 5 ⎠ √ √ √ 3 x y − φ 3x y − 15x y 2 3 3 2 φ 1 4 5

⎛

⎞ x1 y3 + x2 y4 − x3 y1 − x4 y2 = (x1 y1 + x2 y2 + x3 y3 + x4 y4 )1 ⊕ ⎝ −x1 y4 + x2 y3 − x3 y2 + x4 y1 ⎠ x1 y2 − x2 y1 − x3 y4 + x4 y3 3 ⎞ ⎛ x1 y4 + x2 y3 − x3 y2 − x4 y1 ⊕ ⎝ −x1 y3 + x2 y4 + x3 y1 − x4 y2 ⎠ x1 y2 − x2 y1 + x3 y4 − x4 y3 3 √ √ ⎛ ⎞ x√ 5x3 y2 + x4 y1 1 y4 − 5x2 y3 − √ ⎜ − 5x1 y3 + x2 y4 − 5x3 y1 + x4 y2 ⎟ ⎟ √ √ ⊕⎜ ⎝ − 5x1 y2 − 5x2 y1 + x3 y4 + x4 y3 ⎠ x1 y1 + x2 y2 + x3 y3 − 3x4 y4 4

40

4

⎛

2

φ x1 y1 + −√

√ 1 x2 y2 5φ 2

+ x3 y3

AN

⎞

⎟ ⎜ 1 5 ⎜ − √ x1 y2 − √1 x2 y1 − x3 y4 − x4 y3 ⎟ ⎟ ⎜ 5 5 ⎟ ⎜ 1 1 ⊕ ⎜ √5 x1 y3 + x2 y4 + √5 x3 y1 + x4 y2 ⎟ , ⎟ ⎜ ⎜ −x1 y4 − √1 x2 y3 − √1 x3 y2 − x4 y1 ⎟ ⎠ ⎝ 5 5  3 1 − 5 ( φ x1 y1 − φx2 y2 + x3 y3 )

(4.25)

5

⎛

⎞

⎞ y1 x1 ⎜ y2 ⎟ ⎜ ⎟ ⎜ x2 ⎟ ⎜ ⎟ ⊗ ⎜ y3 ⎟ ⎜ ⎟ ⎝ x3 ⎠ ⎝ y4 ⎠ x4 4 y5 5 ⎛

⎛

⎞ √ √ − (φ + 4)x2 y1 + 2φ 2 x3 y4 + 2 5x4 y3 − φ 23 x2 y5 ⎜ ⎟ √ 2 √ 2 2 ⎟ =⎜ 5 − 2φ x2 y2 + φ 2 x3 y3 + 2 5x4 y4 ⎠ ⎝ (φ − 5)x1 y1 + 3φ x1 y√ √ √ 2 2 2φ x1 y3 − φ 2 x2 y4 − 5x3 y1 − 3 3x3 y5 + 2 5x4 y2 2 x y φ2 1 2

3

⎞ √ 1 x y − 3φx1 y5 − φ12 x2 y2 + φ 2 x3 y3 + 5x4 y4 φ2 1 1 √ ⎟ ⎜ √ 3 1 2 2 ⎟ ⊕⎜ ⎝ −φ x1 y2 − φ x2 y1 +√ φ x2 y5 −√φ 2 x3 y4 −√ 5x4 y3 ⎠ 1 x y − φ 2 x2 y4 + 5x3 y1 + 3x3 y5 + 5x4 y2 φ2 1 3 ⎛

√

3

⎛

√ ⎞ √1 x2 y2 + √1 x3 y3 − x4 y4 − √ 3 x1 y5 5 5 5φ  ⎜ ⎟ ⎜ √1 ⎟ 1 3 √1 x3 y4 + x4 y3 ⎟ x y + φx y − ⎜ − 5 x1 y2 + √5φ 2 1 2 5 2 5 5 ⊕⎜ ⎟  ⎜ ⎟ 3 √1 x1 y3 − √1 x2 y4 + x3 y1 − ⎝ ⎠ x y − x y 3 5 4 2 5 5 5 2

φ −√ x1 y1 − 5

−x1 y4 + x2 y3 − x3 y2

⎛ ⎜ ⎜ ⎜ ⎜ ⊕⎜ ⎜ ⎜ ⎜ ⎝ ⎛

1 x y − 3x3 y2 + 3x4 y1 + 53 x4 y5 ) φ2 2 3  φx1 y3 − φ1 x2 y4 − x3 y1 − x4 y2 + 53 x3 y5 1 1 √1 2 φ x1 y2 + φ x2 y1 + 3 φ x2 y5 + φx3 y4 − x4 y3 1 φx1 y1 + √ 2 x1 y5 − φx2 y2 + φ1 x3 y3 − x4 y4 3φ √ √

1 2 2 (φ x1 y4

1 √ (−(φ 2 3

4

−

− 5)x1 y4 + (φ + 4)x2 y3 +

⎞



5x3 y2 −

√ 15 2 x4 y5

5x4 y1 ) + 32 x4 y5

⎟ ⎟ ⎟ ⎟ ⎟ , ⎟ ⎟ ⎟ ⎠ 5

⎞

x1 y4 − x2 y3 − 2x3 y2 + 32 x4 y√1 + ⎟ ⎜ 2 ⎜ φ x1 y3 + 12 x2 y4 − 12 x3 y1 + 215 x3 y5 − x4 y2 ⎟ φ ⎟ ⎜ √ ⎟ ⎜ ⊕ ⎜ − φ12 x1 y2 + 3φ−2 x2 y1 + 23 φx2 y5 + φ 2 x3 y4 − x4 y3 ⎟ , 2 ⎟ ⎜ √ ⎜ 3φ−1 x y − φ 2 x y − 1 x y − x y − 3 x y ⎟ 1 1 2 2 3 3 4 4 1 5 ⎠ ⎝ 2 2 2φ φ √ √ √ 15 3 3x1 y4 + 3x2 y3 + 2 x4 y5 − 2 x4 y1 5

(4.26)

References

41

⎛ ⎞ ⎞ y1 x1 ⎜ y2 ⎟ ⎜ x2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x3 ⎟ ⊗ ⎜ y 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ y4 ⎠ ⎝ x4 ⎠ x5 5 y5 5 = (x1 y1 + x2 y2 + x3 y3 + x4 y4 + x5 y5 )1 √ ⎞ ⎛ x1 y3 − x3 y1 + x2 y4 − x4 y2 + √3(x3 y5 − x5 y3 ) ⊕ ⎝ x1 y4 − x4 y1 − (x2 y3 − x3 y2 ) − 3(x4 y5 − x5 y4 ) ⎠ −2(x1 y2 − x2 y1 ) − (x3 y4 − x4 y3 ) 3 √ ⎞ ⎛ (2φ + 3)(x1 y4 − x4 y1 ) + 2φ(x2 y3 − x3 y2 ) + 4 y5 − x5 y4 ) √ 3(x 2 ⎠ ⊕ ⎝ (φ + 3)(x1 y3 − x3 y1 ) + √2φ(x2 y4 − x4 y2 ) − 3φ (x3 y5 − x5 y3 ) −φ(x1 y2 − x2 y1 ) − 15φ(x2 y5 − x5 y2 ) + 2φ(x3 y4 − x4 y3 )  ⎛ √ 2 ⎞ 3 √ 5( φ2 (x1 y4 + x4 y1 ) + x2 y3 + x3 y2 + 2φ3 (x4 y5 + x5 y4 )) √ ⎜√ ⎟ ⎜ 5( 1 (x1 y3 + x3 y1 ) − (x2 y4 + x4 y2 ) + 3 φ(x3 y5 + x5 y3 )) ⎟ 2 ⎟ 2 2φ ⊕⎜ ⎜ √5 √ ⎟ √ ⎝ (− 5(x1 y2 + x2 y1 ) + 3(x2 y5 + x5 y2 ) + 2(x3 y4 + x4 y3 )) ⎠ 2 3x1 y1 − 2(x2 y2 + x3 y3 + x4 y4 ) + 3x5 y5 4 ⎞ ⎛ 1 1 2 φ √ ( (x1 y4 − x4 y1 ) − (x2 y3 − x3 y2 ) + √ (x4 y5 − x5 y4 )) 2φ 2 3 ⎟ ⎜ 15 φ ⎜ √ ( 2 (x1 y3 − x3 y1 ) − (x2 y4 − x4 y2 ) + √1 2 (x3 y5 − x5 y3 )) ⎟ ⎟ ⎜ 5 2 3φ ⊕⎜ 1 ⎟ 1 1 ⎝ 2√5 (x1 y2 − x2 y1 ) − 2√3 (x2 y5 − x5 y2 ) − √5 (x3 y4 − x4 y3 ) ⎠ − √1 (x1 y5 − x5 y1 ) ⎛

⊕

3

4

√ ⎞ 4 5 1 15 (φx3 y3 + φ x4 y4 ) + x5 y5 ⎟ ⎜ 4 √1 √2 ⎟ ⎜ 3 (x1 y2 + x2 y1 + 15 (x2 y5 + x5 y2 ) + 5 (x3 y4 + x4 y3 )) ⎟ ⎜ 2−3φ ⎟ ⎜ 4 √ √ (−φ(x y + x y ) + 2(x y + x y ) − (x y + x y )) 1 3 3 1 2 4 4 2 3 5 5 3 ⎟ ⎜ 3 5 3 √ √ ⎟ ⎜ 4 5 3 1 ⎟ ⎜ ⎠ ⎝ 15 (− φ (x1 y4 + x4 y1 ) + 2(x2 y3 + x3 y2 ) − √ 3 (3φ − 1)(x4 y5 + x5 y4 )) 4 15 1 √ (−11x1 y1 + 4x2 y2 + 11x5 y5 ) − 45 ((2 − 3φ)x3 y3 + (3φ − 1)x4 y4 ) x 1 y5 + x 5 y1 + 3 15

⎛

−x1 y1 −

11 √ (x y 3 15 1 5

+ x5 y1 ) + 43 x2 y2 −

,

5

√ √ ⎞ √ − 3 4 5 (x1 y1 − x5 y5 ) − 43 (x1 y5 + x5 y1 ) + 5x2 y2 − φ 2 x3 y3 + 12 x4 y4 φ √ √ ⎜ ⎟ ⎜ ⎟ 5(x1 y2 + x2 y1 ) + 3(x2 y5 + x5 y2√ ) + x3 y4 + x4 y3 ⎜ ⎟ ⎜ ⎟ 3 (x y + x y ) 2 −φ (x y + x y ) + x y + x y + ⊕ ⎜ ⎟ . 1 3 3 1 2 4 4 2 3 5 5 3 φ √ ⎜ ⎟ 1 (x y + x y ) + (x y + x y ) − 3φ(x y + x y ) ⎜ ⎟ 1 4 4 1 2 3 3 2 4 5 5 4 2 ⎝ √ ⎠ φ √ √ √ − 43 (x1 y1 − 4x2 y2 − x5 y5 ) + 3 4 5 (x1 y5 + x5 y1 ) + φ3 x3 y3 − 3φx4 y4

⎛

5

(4.27)

References 1. 2. 3. 4.

Shirai, K.: J. Phys. Soc. Jpn. 61, 2735 (1992) Cummins, C.J., Patera, J.: J. Math. Phys. 29, 1736 (1988) Feruglio, F., Paris, A.: J. High Energy Phys. 1103, 101 (2011). arXiv:1101.0393 [hep-ph] Everett, L.L., Stuart, A.J.: Phys. Rev. D 79, 085005 (2009). arXiv:0812.1057 [hep-ph]

Chapter 5

T

In this section, we study the group T  , which is the double covering group of A4 = T . Instead of (4.6) for the case of A4 , we consider the following algebraic relations: s 2 = r,

r 2 = t 3 = (st)3 = e,

rt = tr.

(5.1)

The closed algebra generated by r, s, and t forms the group T  , which contains 24 elements.

5.1 Conjugacy Classes The 24 elements of T  are classified by their orders according to: h=1:

{e},

h=2:

{r},   2 h = 3 : t, t , ts, st, rst 2 , rt 2 s, rtst, rsts ,   h = 4 : s, rs, tst 2 , t 2 st, rtst 2 , rt 2 st ,   h = 6 : rt, rst, rts, rt 2 , sts, st 2 , t 2 s, tst .

(5.2)

Moreover, these elements fall into seven conjugacy classes: C1 : C1 : C4 :

{e}, {r}, {t, rsts, st, ts}, 2   C4 : t , rtst, rt 2 s, rst 2 ,   s, rs, tst 2 , t 2 st, rtst 2 , rt 2 st , C6 : C4 : {rt, sts, rst, rts},   C4 : rt 2 , tst, t 2 s, st 2 ,

h = 1, h = 2, h = 3, h = 3,

(5.3)

h = 4, h = 6, h = 6.

H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_5, © Springer-Verlag Berlin Heidelberg 2012

43

44

5

Table 5.1 Characters of T  representations

h

χ1

χ 1

χ1

χ2

C1

1

1

1

1

2

C1

2

1

1

1

−2

C4

3

1

ω

ω2 ω

T

χ 2

χ2

χ3

2

2

3

−2

−2

3

−1

−ω

−ω2

0

−1

−ω2

−ω

0 0

C4 C4 C4

3

1

ω2

6

1

ω

ω2

1

ω

ω2

6

1

ω2

ω

1

ω2

ω

0

C6

4

1

1

1

0

0

0

−1

5.2 Characters and Representations The orthogonality relations (2.18) and (2.19) for T  lead to m1 + 22 m2 + 32 m3 + · · · = 24,

(5.4)

m1 + m2 + m3 + · · · = 7.

(5.5)

The solution is (m1 , m2 , m3 ) = (3, 3, 1). Therefore, we find three singlets, three doublets, and a triplet in T  . Now consider the characters, which are obtained by a similar analysis to A4 = T . We begin with the singlets. Because s 4 = r 2 = e, there are four possibilities for χα (s) = in (n = 0, 1, 2, 3). However, since t and ts belong to the same conjugacy class, namely C4 , the only value of the character that is consistent with the conjugacy class structure is χα (s) = 1 for the singlets. This also implies that χα (r) = 1. Then, as for A4 = T , the three singlets are classified by three possible values of χα (t) = ωn . That is, the three singlets 1, 1 , and 1 are classified by the three values χα (t) = 1, ω, and ω2 , respectively. These are shown in Table 5.1. Next consider the three doublet representations 2, 2 , and 2 , and the triplet representation 3 for r. The element r commutes with all elements. This implies by Schur’s lemma that r can be represented by   1 0 , (5.6) λ2,2 ,2 0 1 on 2, 2 , and 2 , and

⎛

1 λ3 ⎝ 0 0

0 1 0

⎞ 0 0⎠, 1

(5.7)

on 3. In addition, the possible values of λ2,2 ,2 and λ3 must be equal to λ2,2 ,2 = ±1 and λ3 = ±1 because r 2 = e. Thus, we obtain the possible values of the characters as χ2 (r), χ2 (r), χ2 (r) = ±2 and χ3 (r) = ±3. On the other hand, the second orthogonality relation between e and r gives

χDα (e)∗ χDα (r) = 3 + 2χ2 (r) + 2χ2 (r) + 2χ2 (r) + 3χ3 (r) = 0, (5.8) α

5.2 Characters and Representations

45

where χ1,1 ,1 (r) = 1 has been used. We obtain the solution χ2 (r) = χ2 (r) = χ2 (r) = −2, These results are summarized in Table 5.1. The element r is therefore represented by   1 0 r =− , 0 1 on 2, 2 , and 2 , and

⎛

1 r = ⎝0 0

0 1 0

⎞ 0 0⎠, 1

χ3 (r) = 3.

(5.9)

(5.10)

on 3. We now consider the doublet representation of t. We use the basis diagonalizing t. Because t 3 = e, the element t can be written in the form  k  ω 0 , (5.11) 0 ω with k,  = 0, 1, 2. However, if k = , the above matrix would become proportional to the (2 × 2) identity matrix, that is, the element t would also commute with all the elements. In fact, it is nothing but a singlet representation. Hence, we should have the condition k =  and it follows that there are three possible values for the trace of the above values as ωk + ω = −ωn with k, , n = 0, 1, 2 and k = ,  = n, n = k. That is, the characters of t for the three doublets 2, 2 , and 2 are classified by χ2 (t) = −1, χ2 (t) = −ω, and χ2 (t) = −ω2 . These are shown in Table 5.1. The element t is thus represented by   2 0 ω , on 2, (5.12) t= 0 ω   1 0 t= , on 2 , (5.13) 0 ω2 and

 t=

ω 0

 0 , 1

on 2 .

(5.14)

Since we have found explicit (2 × 2) matrices for r and t on all three doublets, it is straightforward to calculate the explicit forms of rt and rt 2 , which belong to the conjugacy classes C4 and C4 , respectively. Then, it is also straightforward to compute the characters of C4 and C4 for the doublets using the explicit forms of the (2 × 2) matrices for rt and rt 2 . These are shown in Table 5.1. In order to determine the character of t for the triplet χ3 (t), we exploit the second orthogonality relation between e and t:

χDα (e)∗ χDα (t) = 0. (5.15) α

5 T

46

Since all the characters χα (t) except χ3 (t) have been derived in the above, the orthogonality relation (5.15) requires χ3 (t) = 0, that is, χ3 (C4 ) = 0. Similarly, it is found that χ3 (C4 ) = χ3 (C4 ) = χ3 (C4 ) = 0, as shown in Table 5.1. We now consider the explicit form of the (3 × 3) matrix for t on the triplet. We take the basis to diagonalize t. Since t 3 = e and χ3 (t) = 0, we obtain ⎞ ⎛ 1 0 0 (5.16) t = ⎝ 0 ω 0 ⎠ , on 3. 0 0 ω2 Finally, we study the characters of C6 , which contains s, for the doublets and the triplet. Here, we use the first orthogonality relation between the trivial singlet representation and the doublet representation 2:

χ1 (g)∗ χ2 (g) = 0. (5.17) g∈G

Since all the characters except χ2 (C6 ) have already been identified, this orthogonality relation (5.17) requires χ2 (C6 ) = 0. Similarly, we find χ2 (C6 ) = χ2 (C6 ) = 0. In addition, the character  of C6 for the triplet χ3 (C6 ) is also determined using the orthogonality relation g∈G χ1 (g)∗ χ2 (g) = 0 and the known values of the other characters. As a result, we obtain χ3 (C6 ) = −1. We have thus found all characters of the T  group. These are summarized in Table 5.1. Let us now investigate the explicit form of s on the doublets and triplet. On the doublets, this element must act as a (2 × 2) unitary matrix which satisfies tr(s) = 0 and s 2 = r. Recall that the doublet representation for r has already been obtained in (5.9). Thus, the element s could be represented by √   1 i 2p √ s = −√ (5.18) , p = eiφ , 2 p ¯ −i 3 on the doublet representations. For example, for 2, this representation of s satisfies

i (5.19) tr(st) = − √ ω2 − ω = −1, 3 so the ambiguity of p cannot be removed. Similarly, we can study the explicit form of s on the triplet. Here, we summarize the doublet and triplet representations: √   2     1 2p −1 0 ω 0 √i t= on 2, , r= , s = −√ 0 −1 0 ω −i 3 − 2p¯ (5.20) √       1 1 0 i 2p −1 0 √ t= on 2 , , r= , s = −√ 0 −1 0 ω2 −i 3 − 2p¯ (5.21) √       1 ω 0 −1 0 2p √i t= , r= , s = −√ on 2 , 0 1 0 −1 −i 3 − 2p¯ (5.22)

5.3 Tensor Products

47

⎛

⎞ 1 0 0 t = ⎝0 ω 0 ⎠, 0 0 ω2 ⎛ −1 2p1 1 −1 s = ⎝ 2p¯ 1 3 2p¯ p¯ 2p¯ 1 2

⎛

1 r = ⎝0 0 ⎞ 2p1 p2 2p2 ⎠ −1

2

⎞ 0 0 1 0⎠, 0 1

(5.23)

on 3,

where p1 = eiφ1 and p2 = eiφ2 .

5.3 Tensor Products First, we study the tensor product of 2 and 2, i.e.,     x1 y ⊗ 1 . x2 2 y2 2

(5.24)

Let us investigate the transformation properties of elements xi yj , for i, j = 1, 2, under t, r, and s. It is easily found that ⎞ ⎛ i √ p1 p2 p(x       ¯ 1 y2 + x2 y1 ) 2 x1 y2 − x2 y1 y x1 ⎠ . ⊗ 1 = ⊕⎝ √ p2 p¯ 2 x1 y1 x2 2(2 ) y2 2(2 ) 2 1 x2 y2 3 (5.25) Similarly, we obtain 



x1 x2

x1 x2





y ⊗ 1 y 2 2 (2)



 2 (2)

⊗

y1 y2



 2 (2 )

=



x1 y2 − x2 y1 √ 2

 2 (2 )

=

x1 y2 − x2 y1 √ 2

⎛

 1

⊕⎝



√i

2

⎞ p1 p¯ 2 x1 y1 ⎠ , x2 y2 p¯ p¯ 2 (x1 y2 + x2 y1 ) 3

(5.26) ⎞

⎛

x2 y2 i ⊕ ⎝ √2 p¯ p¯ 1 (x1 y2 + x2 y1 ) ⎠ . 1 p¯ 2 p¯ 1 p¯ 2 x1 y1 3 (5.27)

We can also compute other products such as 2 × 2 , 2 × 2 , and 2 × 2 . It is found that 2 × 2 = 2 × 2 ,

2 × 2 = 2 × 2 ,

2 × 2 = 2 × 2.

Moreover, a similar analysis leads to ⎛ ⎞ ⎛ ⎞ x1 y1   ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = x1 y1 + p12 p2 (x2 y3 + x3 y2 ) 1 x3 3 y3 3   ⊕ x3 y3 + p¯ 1 p¯ 22 (x1 y2 + x2 y1 ) 1

(5.28)

5 T

48

  ⊕ (x2 y2 + p¯ 1 p2 (x1 y3 + x3 y1 ) 1 ⎞ ⎛ 2x y − p 2 p (x y + x3 y3 ) 1⎝ 1 12 1 2 2 3 ⊕ 2p1 p2 x3 y3 − x1 y2 − x2 y1 ⎠ 3 2p1 p¯ 2 x2 y2 − x1 y3 − x3 y1 3 ⎞ ⎛ x2 y 3 − x 3 y 2 1 ⊕ ⎝ p¯ 12 p¯ 2 (x1 y2 − x2 y1 ) ⎠ , (5.29) 2 p¯ 12 p¯ 2 (x3 y1 − x1 y3 ) 3 ⎛ ⎞  √    y1 −i x1 2pp1 x2 y2 + x1 y1 ⎝ ⎠ √ ⊗ y2 = x2 2,2 ,2 i 2pp ¯ 1 p2 x1 y3 − x2 y1 2,2 ,2 y3 3   √ −i√ 2pp2 x2 y3 + x1 y2 ⊕ i 2p¯ p¯ 1 x1 y1 − x2 y2 2 ,2 ,2   √ −i √2p p¯ 1 p¯ 2 x2 y1 + x1 y3 ⊕ , (5.30) i 2p¯ p¯ 2 x1 y2 − x2 y3 2 ,2,2     y1 xy1 = , (5.31) (x)1 (1 ) ⊗ y2 2,2 ,2 xy2 2 (2 ),2 (2),2(2 ) ⎛ ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ xy3 xy2 y1 y1 (x)1 ⊗ ⎝ y2 ⎠ = ⎝ p¯ 12 p¯ 2 xy1 ⎠ , (x)1 ⊗ ⎝ y2 ⎠ = ⎝ p¯ 1 p2 xy3 ⎠ . 2 y3 3 y3 3 p¯ 12 p¯ 2 xy1 3 p¯ 1 p¯ 2 xy2 3 (5.32) p

The representations for can be obtained in general by transforming p as follows:    1 0 Φ2 (p), p  = peiγ , Φ2 p = (5.33) 0 e−iγ ⎛ ⎞ 1 0 0  ⎠ Φ3 (p), p1 = p1 eiα , p2 = p2 e−iβ . 0 Φ3 p = ⎝ 0 e−iγ −i(α+β) 0 0 e (5.34) If one takes the parameters p = i and p1 = p2 = 1, then the generator s simplifies to √   i 1 2 √ s = −√ (5.35) , on 2, 2 , 2 , 2 −1 3 ⎛ ⎞ −1 2 2 1⎝ 2 −1 2 ⎠ , on 3. (5.36) s= 3 2 2 −1 These tensor products can also be simplified to: 

x1 x2





y ⊗ 1 y 2 2(2 )



 2(2 )

=

x1 y2 − x2 y1 √ 2



⎛ x1 y2 +x2 y1 ⎞ √

2

⊕ ⎝ −x1 y1 ⎠ , 1 x2 y2 3

(5.37)

5.3 Tensor Products





x1 x2

x1 x2





y ⊗ 1 y 2 2 (2)



 2 (2)

⊗

y1 y2

49



 2 (2 )

=



 2 (2 )

=

x1 y2 − x2 y1 √ 2 x1 y2 − x2 y1 √ 2

 1

 1

⎛

⎞ −x1 y1 ⊕ ⎝ x2 y2 ⎠ , ⎛ ⊕⎝

x1 y2√ +x2 y1 2

x2 y2

x1 y2√ +x2 y1 2 −x1 y1

⎞

(5.38)

3

⎠ ,

(5.39)

3 ⎛ ⎞ ⎞ y1 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = [x1 y1 + x2 y3 + x3 y2 ]1 x3 3 y3 3 ⊕ [x3 y3 + x1 y2 + x2 y1 ]1 ⊕ [x2 y2 + x1 y3 + x3 y1 ]1 ⎛ ⎞ 2x y − x2 y3 − x3 y2 1⎝ 1 1 2x3 y3 − x1 y2 − x2 y1 ⎠ ⊕ 3 2x y − x y − x y 2 2 1 3 3 1 3 ⎛ ⎞ x2 y3 − x3 y2 1 ⊕ ⎝ x1 y2 − x2 y1 ⎠ , (5.40) 2 x y −x y 3 1 1 3 3 ⎛ ⎞     √ √ y1 2x2 y2 + x1 y1 2x2 y3 + x1 y2 x1 ⎝ ⎠ √ √ ⊗ y2 = ⊕ x2 2,2 ,2 2x1 y3 − x2 y1 2,2 ,2 2x1 y1 − x2 y2 2 ,2 ,2 y3 3 √  2x2 y1 + x1 y3 √ ⊕ , (5.41) 2x1 y2 − x2 y3 2 ,2,2     y xy1 (x)1 (1 ) ⊗ 1 = , (5.42) y2 2,2 ,2 xy2 2 (2 ),2 (2),2(2 ) ⎛ ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ y1 xy3 y1 xy2 (x)1 ⊗ ⎝ y2 ⎠ = ⎝ xy1 ⎠ , (x)1 ⊗ ⎝ y2 ⎠ = ⎝ xy3 ⎠ . (5.43) y3 3 xy2 3 y3 3 xy1 3

⎛

When p = eiπ/12 , and p1 = p2 = ω, the representations and their tensor products are as given in Appendix E.2.

Chapter 6

DN

In this chapter, we discuss the dihedral group, which is denoted by DN . It is the symmetry group of the regular polygon with N sides. This group is isomorphic to ZN  Z2 and is also denoted by Δ(2N ). It consists of cyclic rotations ZN and reflections. That is, it is generated by two generators a and b, which act on the N edges xi (i = 1, . . . , N ) of the N -polygon according to a : (x1 , x2 , . . . , xN ) → (xN , x1 , . . . , xN−1 ),

(6.1)

b : (x1 , x2 , . . . , xN ) → (x1 , xN , . . . , x2 ).

(6.2)

These two generators satisfy a N = e,

bab = a −1 ,

b2 = e,

(6.3)

where the third equation is equivalent to aba = b. The order of DN is 2N , and all of the 2N elements can be written in the form a m bk , with m = 0, . . . , N − 1 and k = 0, 1. The third equation in (6.3) implies that the ZN subgroup including a m is a normal subgroup of DN . Thus, DN corresponds to a semi-direct product between ZN including a m and Z2 including bk , i.e., ZN  Z2 . Equation (6.1) corresponds to the (reducible) N -dimensional representation. The simple doublet representation is  a=

cos 2π/N sin 2π/N

− sin 2π/N cos 2π/N



 ,

b=

 1 0 . 0 −1

(6.4)

6.1 DN with N Even The groups DN have different features for N even and odd. We begin by studying DN when N is even. H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_6, © Springer-Verlag Berlin Heidelberg 2012

51

52

6

DN

6.1.1 Conjugacy Classes The algebraic relations (6.3) tell us that a m and a N−m belong to the same conjugacy class and also that b and a 2m b belong to the same conjugacy class. When N is even, DN has the following 3 + N/2 conjugacy classes: C1 : (1)

{e},   a, a N−1 ,

C2 :

h = 1, h = N,

.. .

.. .   (N/2−1) C2 : a N/2−1 , a N/2+1 ,  N/2  , a C1 :   CN/2 : b, a 2 b, . . . , a N−2 b ,    : ab, a 3 b, . . . , a N−1 b , CN/2

.. . h = N/gcd(N, N/2 − 1),

(6.5)

h = 2, h = 2, h = 2,

where we have also shown the order h of each element in the conjugacy class and gcd stands for greatest common divisor. This implies that there are 3 + N/2 irreducible representations. Furthermore, the orthogonality relation (2.18) requires 

2  χα (C1 ) = mn n2 = m1 + 4m2 + 9m3 + · · · = 2N,

α

(6.6)

n

for the mi , which also satisfy m1 + m2 + m3 + · · · = 3 + N/2. The solution is found to be (m1 , m2 ) = (4, N/2 − 1). Therefore, it is found that there are four singlets and (N/2 − 1) doublets.

6.1.2 Characters and Representations We start by studying singlets when N is even, in which case there are four singlets.  Since the generators satisfy b2 = e in CN/2 and (ab)2 = e in CN/2 , the characters  ) = ±1. Thereχα (g) for the four singlets should be χα (CN/2 ) = ±1 and χα (CN/2  fore, there are four possible combinations of χα (CN/2 ) = ±1 and χα (CN/2 ) = ±1 and they correspond to four singlets, 1±± , which are presented in Table 6.1. Now consider the doublet representations, that is, (2 × 2) matrix representations. As can be seen from (6.4), these (2 × 2) matrices correspond to one of the doublet representations. The (2 × 2) matrix representations for the generic doublet 2k are obtained by replacing a → ak .

(6.7)

6.1

DN with N Even

53

Table 6.1 Characters of DN even representations h

χ1++ χ1+−

χ1−+

C1

1

1

1

1

1

2

C2(1) .. .

N

1

−1

−1

1

2 cos(2πk/N)

1

2 cos[2πk(N/2 − 1)/N]

1

−2

(N/2−1)

χ1−− χ2k

N/gcd(N, N/2 − 1) 1

(−1)(N/2−1) (−1)(N/2−1)

C1

2

1

(−1)N/2

(−1)N/2

CN/2

2

1

1

−1

−1

0

 CN/2

2

1

−1

1

−1

0

C2

Thus, a and b are represented for the doublet 2k by   cos 2πk/N − sin 2πk/N a= , sin 2πk/N cos 2πk/N

 b=

 1 0 , 0 −1

(6.8)

where k = 1, . . . , N/2 − 1 for N even and k = 1, . . . , (N − 1)/2 for N = odd. In the expression for the doublet 2k by   xk 2k = , (6.9) yk the generator a is the ZN rotation on the two-dimensional real coordinates (xk , yk ), and the generator b is the reflection along yk , i.e., yk → −yk . These transformations can be represented on the complex coordinate zk and its conjugate z¯ −k . The bases transform according to       1 1 i xk zk =U , U=√ . (6.10) z¯ −k yk 2 1 −i In the complex basis, the generators a and b can be expressed by a˜ = U aU −1 and b˜ = U bU −1 :     exp(2πik/N ) 0 0 1 ˜ a˜ = , b= . (6.11) 1 0 0 exp(−2πik/N ) This complex basis may be useful. Actually, the generator a˜ is the diagonal matrix. This implies that, in the doublet 2k , which is denoted by   zk 2k = , (6.12) z¯ −k each of the up and down components zk and z¯ −k , respectively, has definite ZN charge. Indeed, the ZN charges of zk and z¯ −k are equal to k and −k, respectively.

54

6

DN

The characters of these matrices for the doublets 2k are presented in Table 6.1. It is easy to show that these characters satisfy the orthogonality relations (2.16) and (2.17).

6.1.3 Tensor Products In the next step, we discuss the tensor products of the group DN with N even. Let us start with 2k × 2k , i.e.,     zk zk ⊗ , (6.13) z¯ −k 2 z¯ −k  2  k

k

k, k 

where = 1, . . . , N/2 − 1. Note that zk zk , zk z¯ −k  , z¯ −k zk , and z¯ −k z¯ −k  have definite ZN charges, i.e., k + k  , k − k  , −k + k  , and −k − k  , respectively. For the case with k + k  = N/2 and k − k  = 0, they are decomposed into two doublets as         zk zk zk zk  z z¯  ⊗ = ⊕ k −k . (6.14) z¯ −k 2 z¯ −k  2  z¯ −k z¯ −k  2  z¯ −k zk  2  k

k

k+k

k−k

In the case of k + k  = N/2, the matrix a is represented on the above (reducible) doublet (zk zk  , z¯ −k z¯ −k  ) by      zk zk  −1 0 zk zk  a = . (6.15) z¯ −k z¯ −k  z¯ −k z¯ −k  0 −1 Since a is proportional to the (2 × 2) identity matrix for (zk zk , z¯ −k z¯ −k  ) with k + k  = N/2, we can diagonalize another matrix b in this vector space (zk zk  , z¯ −k z¯ −k  ). Such a basis is (zk zk + z¯ −k z¯ −k  , zk zk  − z¯ −k z¯ −k  ), and the eigenvalues of b are      z z  + z¯ z¯   k k −k −k zk zk  + z¯ −k z¯ −k  1 0 b . (6.16) = zk zk  − z¯ −k z¯ −k  0 −1 zk zk  − z¯ −k z¯ −k  Thus, zk zk  + z¯ −k z¯ −k  and zk zk  − z¯ −k z¯ −k  correspond to 1+− and 1−+ , respectively. When k − k  = 0, a similar decomposition is obtained for the (reducible) doublet (zk z¯ −k  , z¯ −k zk ). The generator a is the (2 × 2) identity matrix on the vector space (zk z¯ −k  , z¯ −k zk ) with k − k  = 0. Therefore, we can take the basis (zk z¯ −k  + z¯ −k zk , zk z¯ −k  − z¯ −k zk  ), where b is diagonalized. That is, zk z¯ −k  + z¯ −k zk  and zk z¯ −k  − z¯ −k zk correspond to 1++ and 1−− , respectively. Now, we study the tensor products of the doublets 2k and singlets, for example, 1−− × 2k . Here we denote the vector space for the singlet 1−− by w, where aw = w and bw = −w. It is easily found that (wzk , −w z¯ k ) is nothing but the doublet 2k , that

6.1

DN with N Even

55

is, 1−− × 2k = 2k . Similar results are obtained for other singlets. Furthermore, it is straightforward to study the tensor products among singlets. Hence, the tensor products of DN irreducible representations with N even can be summarized as follows:         zk zk zk zk z¯ −k  zk ⊗ = ⊕ , (6.17) z¯ −k 2 z¯ −k  2  z¯ −k z¯ −k  2  z¯ −k zk  2  k

k

k+k

k−k

for k + k  = N/2 and k − k  = 0,     zk zk  ⊗ = (zk zk  + z¯ −k z¯ −k  )1+− ⊕ (zk zk  − z¯ −k z¯ −k  )1−+ z¯ −k 2 z¯ −k  2  k k   zk z¯ −k  ⊕ , (6.18) z¯ −k zk  2  k−k

for k + k  = N/2 and k − k  = 0,     zk zk  ⊗ = (zk z¯ −k  + z¯ −k zk  )1++ ⊕ (zk z¯ −k  − z¯ −k zk  )1−− z¯ −k 2 z¯ −k  2  k k   zk zk  ⊕ , (6.19) z¯ −k z¯ −k  2  k+k

for k + k  = N/2 and k − k  = 0,     zk zk  ⊗ = (zk z¯ −k  + z¯ −k zk  )1++ ⊕ (zk z¯ −k  − z¯ −k zk  )1−− z¯ −k 2 z¯ −k  2  k

k

⊕(zk zk + z¯ −k z¯ −k  )1+− ⊕ (zk zk − z¯ −k z¯ −k  )1−+ , (6.20) for k + k  = N/2 and k − k  = 0, and   zk (w)1++ ⊗ z¯ −k 2 k   zk (w)1−− ⊗ z¯ −k 2 k   zk (w)1+− ⊗ z¯ −k 2 k   zk (w)1−+ ⊗ z¯ −k 2 k

 =

wzk w z¯ −k

 , 2k

 wzk , −w z¯ −k 2 k   w z¯ −k = , wzk 2 k   w z¯ −k = , −wzk 2 

=

1s1 s2 ⊗ 1s1 s2 = 1s1 s2 ,

(6.21)

k

(6.22)

56

6

DN

with si , si , si = ± (i = 1, 2), where si = + for (si , si ) = (+, +) and (−, −), and si = − for (si , si ) = (+, −) and (−, +). Hereafter, this sign rule for si will be denoted by si = si si (i = 1, 2) for simplicity. Note that the above multiplication rules are the same for the complex basis and the real basis. For example, in both bases we get 2k ⊗ 2k = 2k+k + 2k−k , for k + k  = N/2 and k − k  = 0. On the other hand, the elements of the doublets are written in a different way, although they transform according to (6.10).

6.2 DN with N Odd We now consider DN with N odd, carrying out a similar study of the conjugacy classes, characters, representations, and tensor products.

6.2.1 Conjugacy Classes The group DN with N odd has the following 2 + (N − 1)/2 conjugacy classes: C1 : (1) C2 :

.. . ((N−1)/2)

C2

CN :

:

{e},   a, a N−1 ,

h = 1,

.. .  (N−1)/2 (N+1)/2  , a ,a   N−1 b, ab, . . . , a b ,

.. .

h = N, (6.23)

 h = N/gcd N, (N − 1)/2 , h = 2.

That is, there are 2 + (N − 1)/2 irreducible representations. Furthermore, the orthogonality relation (2.18) requires the same equation as (6.6) for the mi , which also satisfy m1 + m2 + m3 + · · · = 2 + (N − 1)/2. The solution is found to be (m1 , m2 ) = (2, (N − 1)/2). Thus, there are two singlets and (N − 1)/2 doublets.

6.2.2 Characters and Representations We study the two singlets of DN with N odd. Since b2 = e holds in CN , the characters χα (g) for the two singlets should be χα (CN ) = ±1. Since both b and ab belong to the same conjugacy class CN , the characters χα (a) for the two singlets

6.2

DN with N Odd

57

Table 6.2 Characters of DN =odd representations h

χ1 +

C1

1

1

1

2

C2(1) .. .

N

1

1

2 cos(2πk/N)

C2

N/gcd(N, (N − 1)/2)

1

1

2 cos[2πk(N − 1)/2N]

CN

2

1

−1

((N −1)/2)

χ 1−

χ2k

0

must always satisfy χα (a) = 1. That is, there are two singlets 1+ and 1− . Their characters are determined by whether the conjugacy class includes b or not, as shown in Table 6.2. The doublet representations of DN with N odd are the same as those in DN with N even. Their characters are also shown in Table 6.2.

6.2.3 Tensor Products Let us discuss the tensor products of the irreducible representations of DN with N odd. We can analyze them in a similar way to those of DN with N even. The results are summarized as follows:         zk zk zk zk z z¯  ⊗ = ⊕ k −k , (6.24) z¯ −k 2 z¯ −k  2  z¯ −k z¯ −k  2  z¯ −k zk  2  k

k

k+k

k−k

for k − k  = 0, where k, k  = 1, . . . , N/2 − 1,     zk  zk ⊗ = (zk z¯ −k  + z¯ −k zk )1+ ⊕ (zk z¯ −k  − z¯ −k zk  )1− z¯ −k 2 z¯ −k  2  k k   zk zk  ⊕ , (6.25) z¯ −k z¯ −k  2  k+k

for k − k  = 0, and



   zk wzk = , z¯ −k 2 w z¯ −k 2 k k     zk wzk (w)1− ⊗ = , z¯ −k 2 −w z¯ −k 2 (w)1++ ⊗

k

1s ⊗ 1s  = 1s  , where s  = ss  .

(6.26)

k

(6.27)

58

6

DN

Fig. 6.1 The D4 symmetry group of a square

6.3 D4 In this and the next section, we present simple examples of DN . The smallest nonAbelian group DN is D3 . However, D3 corresponds to the group of all possible permutations of three objects, that is, it is just S3 . We thus examine D4 and D5 as simple examples. The group D4 is the symmetry group of a square, which is generated by the π/2 rotation a and the reflection b. These satisfy a 4 = e, b2 = e, and bab = a −1 (see Fig. 6.1). D4 thus consists of the eight elements a m bk with m = 0, 1, 2, 3 and k = 0, 1. D4 has the following five conjugacy classes, C1 : C2 : C1 : C2 : C2

:

{e},   a, a 3 ,  2 a ,   b, a 2 b ,   ab, a 3 b ,

h = 1, h = 4, h = 2,

(6.28)

h = 2, h = 2,

where h is the order of each element in the conjugacy class. D4 has four singlets 1++ , 1+− , 1−+ , and 1−− , and one doublet 2. The characters are shown in Table 6.3. The tensor products are:       z z ⊗  = z¯z + z¯ z 1 ⊕ z¯z − z¯ z 1 ++ −− z¯ 2 z¯ 2    ⊕ zz + z¯ z¯  1 ⊕ zz − z¯ z¯  1 , (6.29) +−

    z wz (w)1++ ⊗ = , z¯ 2 w z¯ 2     z w z¯ (w)1+− ⊗ = , z¯ 2 wz 2

−+

    z wz (w)1−− ⊗ = , z¯ 2 −w z¯ 2     z w z¯ (w)1−+ ⊗ = , z¯ 2 −wz 2

(6.30)

6.4

59

D5

Table 6.3 Characters of D4 representations

h

χ1++

χ1+−

χ1−+

χ1−−

χ2

C1

1

1

C2

4

1

1

1

1

2

−1

−1

1

C1 C2 C2

0

2

1

1

2

1

1

1

1

−2

−1

−1

2

1

−1

1

0

−1

0

Fig. 6.2 The D5 symmetry group of a regular pentagon

1s1 s2 ⊗ 1s1 s2 = 1s1 s2 ,

(6.31)

where s1 = s1 s1 and s2 = s2 s2 .

6.4 D5 The group D5 is the symmetry group of a regular pentagon. This is generated by the 2π/5 rotation a and the reflection b (see Fig. 6.2). The generators satisfy a 5 = e, b2 = e, and bab = a −1 . D5 thus contains the 10 elements a m bk with m = 0, 1, 2, 3, 4 and k = 0, 1. They are classified into the following four conjugacy classes: C1 : (1) C2 :

C2(2) : C5 :

{e},   a, a 4 ,  2 3 a ,a ,   b, ab, a 2 b, a 3 b, a 4 b ,

h = 1, h = 5, h = 5,

(6.32)

h = 2.

D5 has two singlets 1+ and 1− , and two doublets 21 and 22 . Their characters are shown in Table 6.4.

60

6

Table 6.4 Characters of D5 representations

χ 1−

χ 21

h

χ1+

C1

1

1

1

2

2

C2(1)

5

1

1

2 cos(2π/5)

2 cos(4π/5)

C2(2)

5

1

1

2 cos(4π/5)

2 cos(8π/5)

C5

2

1

−1

0

0

The tensor products are         z zz z¯z z ⊗  = ⊕ , z¯ 2 z¯ z¯  2 z¯ z 2 z¯ 2 2 1 2 1        z zk ⊗ k = zk z¯ −k + z¯ −k zk 1 + z¯ −k 2 z¯ −k 2 k k     zk zk  ⊕ zk z¯ −k − z¯ −k zk 1 ⊕ ,  − z¯ −k z¯ −k 22k     zk wzk = , (w)1+ ⊗ z¯ −k 2 w z¯ −k 2 k k     zk wzk = , (w)1− ⊗ z¯ −k 2 −w z¯ −k 2 k

1s ⊗ 1s  = 1s  , where s  = ss  .

DN

χ2 2

(6.33)

(6.34)

(6.35)

k

(6.36)

Chapter 7

QN

The binary dihedral group is denoted by QN , where N is even. It consists of the elements a m bk with m = 0, . . . , N − 1 and k = 0, 1, where the generators a and b satisfy a N = e,

b2 = a N/2 ,

b−1 ab = a −1 .

(7.1)

The order of QN is 2N . The generator a can be represented by the same (2 × 2) matrix as for DN , i.e.,   exp(2πik/N ) 0 a= . (7.2) 0 exp(−2πik/N ) Note that a N/2 = e for k even and a N/2 = −e for k odd. This leads to b2 = e for k even and b2 = −e for k odd. Thus, the generators a and b are represented by (2 × 2) matrices, e.g.,     exp(2πik/N ) 0 0 i a= , b= , (7.3) i 0 0 exp(−2πik/N ) for k = odd, a=



exp(2πik/N )

0

0

exp(−2πik/N )



 ,

b=

 0 1 , 1 0

(7.4)

for k = even.

7.1 QN with N = 4n The QN groups have different features between N = 4n and 4n + 2. We begin by investigating QN with N = 4n. H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_7, © Springer-Verlag Berlin Heidelberg 2012

61

62

7

QN

7.1.1 Conjugacy Classes The conjugacy classes are given by the algebraic relations (7.1). The elements are classified into the following (3 + N/2) conjugacy classes: C1 :

{e},   a, a N −1 ,

C2(1) : .. . (N/2−1)

C2

C1 : CN/2 :  : CN/2

:

h = 1, h = N,

.. . 

.. .

 a N/2−1 , a N/2+1 ,  N/2  , a   2 b, a b, . . . , a N −2 B ,   ab, a 3 b, . . . , a N −1 B ,

h = N/ gcd(N, N/2 − 1),

(7.5)

h = 2, h = 4, h = 4,

where h is the order of each element in the given conjugacy class. These are almost the same as the conjugacy classes of DN with N even. There must be (3 + N/2) irreducible representations, and similarly to DN even , there are four singlets and (N/2 − 1) doublets.

7.1.2 Characters and Representations The characters of QN for doublets are the same as those of DN even , and are shown in Table 7.1. We study the characters of the singlets of QN with N = 4n. We then have the relation b2 = a 2n .

(7.6)

Since b4 = e holds in CN/2 , the characters χα (b) for the four singlets must be χα (b) = eπin/2 with n = 0, 1, 2, 3. Furthermore, note that the element ba 2 belongs to the same conjugacy class as b. That implies χα (a 2 ) = 1 for the four singlets. Using (7.6), we have χα (b2 ) = 1, that is, χα (b) = ±1. Thus, the characters for the singlets of QN with N = 4n are the same as those of DN even , as shown in Table 7.1.

7.1.3 Tensor Products The tensor products of the irreducible representations of QN can be analyzed in a similar way to those of DN with N even. The results for QN with N = 4n are:

7.1 QN with N = 4n

63

Table 7.1 Characters of QN representations for N = 4n h

χ1++ χ1+−

χ1−+

C1

1

1

1

1

1

2

C2(1) .. .

N

1

−1

−1

1

2 cos(2πk/N)

1

2 cos[2πk(N/2 − 1)/N]

1

−2

(N/2−1)

χ1−− χ2k

N/ gcd(N, N/2 − 1) 1

(−1)(N/2−1) (−1)(N/2−1)

C1

2

1

(−1)N/2

(−1)N/2

CN/2

4

1

1

−1

−1

0

 CN/2

4

1

−1

1

−1

0

C2



zk z¯ −k



 ⊗ 2k

zk  z¯ −k 

 2k 

 zk zk   (−1)kk z¯ −k z¯ −k  2  k+k   zk z¯ −k  ⊕ ,  (−1)kk z¯ −k zk  2 

 =

(7.7)

k−k

for k + k  = N/2 and k − k  = 0,       zk  zk ⊗ = zk zk  + (−1)kk z¯ −k z¯ −k  1 +− z¯ −k 2 z¯ −k  2  k

k

  ⊕ zk zk  − (−1)kk z¯ −k z¯ −k  1 −+   zk z¯ −k  ⊕ ,  (−1)kk z¯ −k zk  2 

(7.8)

k−k

for k + k  = N/2 and k − k  = 0,       zk zk  ⊗ = zk z¯ −k  + (−1)kk z¯ −k zk  1 ++ z¯ −k 2 z¯ −k  2  k

k

  ⊕ zk z¯ −k  − (−1)kk z¯ −k zk  1 −−   zk zk  ⊕ ,  kk (−1) z¯ −k z¯ −k  2  k+k

for k + k  = N/2 and k − k  = 0,       zk zk ⊗ = zk z¯ −k  + (−1)kk z¯ −k zk  1 ++ z¯ −k 2 z¯ −k  2  k

k

  ⊕ zk z¯ −k  − (−1)kk z¯ −k zk 1

−−

(7.9)

64

7

  ⊕ zk zk + (−1)kk z¯ −k z¯ −k  1 +−   ⊕ zk zk − (−1)kk z¯ −k z¯ −k  1 , −+

for k + k  = N/2 and k − k  = 0,     zk wzk (w)1++ ⊗ = , z¯ −k 2 w z¯ −k 2 k k     zk w z¯ −k = , (w)1+− ⊗ z¯ −k 2 wzk 2 k

k

QN

(7.10)

   zk wzk (w)1−− ⊗ = , z¯ −k 2 −w z¯ −k 2 k k     zk w z¯ −k (w)1−+ ⊗ = , z¯ −k 2 −wzk 2 k k (7.11) 

1s1 s2 ⊗ 1s1 s2 = 1s1 s2 ,

(7.12)

where s1 = s1 s1 and s2 = s2 s2 . It should be noted that some minus signs differ from those occurring in the tensor products for DN .

7.2 QN with N = 4n + 2 As for QN with N = 4n, we now investigate QN with N = 4n + 2.

7.2.1 Conjugacy Classes The conjugacy classes of QN with N = 4n + 2 are exactly the same as those of QN with N = 4n.

7.2.2 Characters and Representations The characters of QN for doublets are the same as those of DN even , and are shown in Table 7.2. Let us consider the four singlets of QN for N = 4n + 2. In this case we have the relation b2 = a 2n+1 .

(7.13)

Since b and a 2 b are in the same conjugacy class, the characters χα (a 2 ) for the four singlets must be χα (a 2 ) = 1. Thus, we obtain χα (a) = ±1. When χα (a) = 1, the relation (7.13) leads to the two possibilities χα (b) = ±1. On the other hand, when χα (a) = −1, the relation (7.13) gives the alternative possibilities χα (b) = ±i. Thus, there are four possibilities corresponding to the four singlets. Note also that χα (a) = χα (b2 ) is satisfied for all the singlets.

7.2 QN with N = 4n + 2

65

Table 7.2 Characters of QN representations for N = 4n + 2 h

χ1++ χ1+−

χ1−+

C1

1

1

1

1

1

2

C2(1) .. .

N

1

−1

−1

1

2 cos(2πk/N)

1

2 cos[2πk(N/2 − 1)/N]

1

−2

(N/2−1)

χ1−− χ2k

N/ gcd(N, N/2 − 1) 1

(−1)(N/2−1) (−1)(N/2−1)

C1

2

1

(−1)N/2

(−1)N/2

CN/2

4

1

i

−i

−1

0

 CN/2

4

1

−i

i

−1

0

C2

7.2.3 Tensor Products We can obtain the tensor products of representations of QN with N = 4n + 2 in a similar way to those of QN with N = 4n, as follows:         zk zk  zk z¯ −k  zk  zk ⊗ = ⊕ ,   z¯ −k 2 z¯ −k  2  (−1)kk z¯ −k z¯ −k  2  (−1)kk z¯ −k zk  2  k k

k+k

k−k

(7.14) for k + k  = N/2 and k − k  = 0,     zk zk  ⊗ = (zk zk  + z¯ −k z¯ −k  )1+− z¯ −k 2 z¯ −k  2  k

k

⊕ (zk zk − z¯ −k z¯ −k  )1−+   zk z¯ −k  ⊕ ,  (−1)kk z¯ −k zk  2 

(7.15)

k−k

for k + k  = N/2 and k − k  = 0,       zk zk  ⊗ = zk z¯ −k  + (−1)kk z¯ −k zk  1 ++ z¯ −k 2 z¯ −k  2  k

k

  ⊕ zk z¯ −k  − (−1)kk z¯ −k zk  1 −−   zk zk  ⊕ ,  (−1)kk z¯ −k z¯ −k  2  k+k

for k

+ k

− k

= N/2 and k = 0,       zk zk  ⊗ = zk z¯ −k  + (−1)kk z¯ −k zk  1 ++  z¯ −k 2 z¯ −k 2  k

k

(7.16)

66

7

QN

  ⊕ zk z¯ −k  − (−1)kk z¯ −k zk  1

−−

⊕ (zk zk + z¯ −k z¯ −k  )1+− ⊕ (zk zk − z¯ −k z¯ −k  )1−+ ,

(7.17)

for k + k  = N/2 and k − k  = 0,  (w)1++ ⊗  (w)1+− ⊗

zk z¯ −k zk z¯ −k



 = 

2k

 =

2k

wzk w z¯ −k w z¯ −k wzk



 , 

2k

, 2k



 wzk , −w z¯ −k 2 2k k     zk w z¯ −k (w)1−+ ⊗ = , z¯ −k 2 −wzk 2 k k (7.18) (w)1−− ⊗

zk z¯ −k



=

1s1 s2 ⊗ 1s1 s2 = 1s1 s2 ,

(7.19)

where s1 = s1 s1 and s2 = s2 s2 .

7.3 Q4 Simple examples of QN are useful for applications. Here we present the results for Q4 and in the next section we discuss Q6 . The group Q4 contains the eight elements a m bk , for m = 0, 1, 2, 3 and k = 0, 1, where a and b satisfy a 4 = e, b2 = a 2 , and b−1 ab = a −1 . These elements are classified into five conjugacy classes: C1 : C2 : C1 : C2 : C2 :

{e},   a, a 3 ,  2 a ,   b, a 2 b ,   ab, a 3 b ,

h = 1, h = 4, h = 2,

(7.20)

h = 4, h = 4.

Q4 has four singlets 1++ , 1+− , 1−+ , and 1−− , and one doublet 2. The characters are shown in Table 7.3. The tensor products are:       z z ⊗  = z¯z − z¯ z 1 ⊕ z¯z + z¯ z 1 ++ −− z¯ 2 z¯ 2    ⊕ zz − z¯ z¯  1 ⊕ zz + z¯ z¯  1 +−

−+

,

(7.21)

7.4 Q6

67

Table 7.3 Characters of Q4 representations

h

χ1++

C1

1

1

C2

4

1

C1 C2 C2

2

1

1

4

1

1

4

1

−1

1

    z wz (w)1++ ⊗ = , z¯ 2 w z¯ 2     z w z¯ (w)1+− ⊗ = , z¯ 2 wz 2

χ1+−

χ1−+

χ1−−

χ2

1

1

1

2

−1

−1

1

2 cos(π/2)

1

1

−2

−1

−1

0

−1

0

    z wz (w)1−− ⊗ = , z¯ 2 −w z¯ 2     z w z¯ (w)1−+ ⊗ = , z¯ 2 −wz 2

1s1 s2 ⊗ 1s1 s2 = 1s1 s2 ,

(7.22)

(7.23)

where s1 = s1 s1 and s2 = s2 s2 . Some minus signs differ from those occurring in the tensor products for D4 .

7.4 Q6 The group Q6 has 12 elements a m bk , for m = 0, 1, 2, 3, 4, 5 and k = 0, 1, where a and b satisfy a 6 = e, b2 = a 3 , and b−1 ab = a −1 . These elements are classified into six conjugacy classes: C1 : (1) C2 : (2) C2 : C1 :

C3 : C3 :

{e},   a, a 5 ,  2 4 a ,a ,  3 a ,   b, a 2 b, a 4 b ,   ab, a 3 b, a 5 b ,

h = 1, h = 6, h = 3,

(7.24)

h = 2, h = 4, h = 4.

Q6 has four singlets 1++ , 1+− , 1−+ , and 1−− , and two doublets 21 and 22 . The characters are shown in Table 7.4. The tensor products are:         z z z¯z ⊗  = zz − z¯ z¯  1 ⊕ zz + z¯ z¯  1 ⊕ , (7.25) +− −+ z¯ 2 z¯ 2 z¯ z 2 2

1

1

68

7

Table 7.4 Characters of Q6 representations

h χ1++ χ1+−

χ1−+

χ1−− χ21

QN

χ2 2

1 1

1

1

1

2

C2(1) 6 1

−1

−1

1

2 cos(2π/6) 2 cos(4π/6)

C2(2) 3 1

1

1

1

2 cos(4π/6) 2 cos(8π/6)

C1

2 1

1

1

1

−2

2

C3

4 1

i

−i

−1

0

0

C3

4 1

−i

i

−1

0

0

C1

2

         z z zz ⊗  = z¯z − z¯ z 1 ⊕ z¯z + z¯ z 1 ⊕ , (7.26) ++ −− z¯ 2 z¯ 2 −¯zz¯  2  k

k

k

for k, k  = 1, 2 and k  = k, and 

 zk (w)1++ ⊗ z¯ −k 2 k   zk (w)1−− ⊗ z¯ −k 2 k   zk (w)1+− ⊗ z¯ −k 2 k   zk (w)1−+ ⊗ z¯ −k 2 k

 = 

wzk w z¯ −k

, 2k

 wzk = , −w z¯ −k 2 k   w z¯ −k = , wzk 2 k   w z¯ −k = , −wzk 2

1s1 s2 ⊗ 1s1 s2 = 1s1 s2 , where s1 = s1 s1 and s2 = s2 s2 .



(7.27)

k

(7.28)

Chapter 8

QD2N

8.1 Generic Aspects Here, we briefly study generic aspects of QD2N . Let us start with the semi-direct  product Z2N  −1  Z2 , which has order 2N . The Abelian groups Z2N  −1 and Z2 are generated by two generators a and b, respectively, which satisfy N  −1

a2

= 1,

b2 = 1.

(8.1)

Because of the semi-direct product, we require bab−1 = a m .

(8.2)

N  −1

If m = 1 mod 2 , this corresponds to the direct product. Thus, we require m = 1  mod 2N −1 . Furthermore, using a = ba m b, it is found that     2 a m = ba m b · · · ba m b = ba m b. (8.3) 2

Note that a m = bab, because a = ba m b. Then consistency requires a m = a. This implies 

m2 = 1 mod 2N −1 .

(8.4)

It is generally known that the solutions of the above equation are m = ±1 for N  ≤ 3  and m = ±1, 2N −2 ± 1 for N  ≥ 4. For m = 1, the result is nothing but a direct product group. For m = −1, one finds  that it is identified as a dihedral group. For the non-trivial solution m = 2N −2 − 1, the group Z2N  −1  Z2 is called the quasi-dihedral group.1 Therefore, we define QD2N  for N  ≥ 4, e.g., QD16 , QD32 , etc.  Hereafter we use the notation N ≡ 2N −1 for convenience. Then, the group QD2N is isomorphic to ZN  Z2 and the generators a and b of ZN and Z2 , respectively, 1 No



name has been adopted for the non-trivial solution m = 2N −2 + 1.

H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_8, © Springer-Verlag Berlin Heidelberg 2012

69

70

8

QD2N

satisfy a N = 1,

b2 = 1,

bab = a N/2−1 .

(8.5)

for k = 0, . . . , N − 1 and All elements of QD2N can be written in the form  = 0, 1. The generators a and b are represented, e.g., by     ρ 0 0 1 , (8.6) , b= a= 1 0 0 ρ N/2−1 a k b

where ρ = e2πi/N .

8.1.1 Conjugacy Classes The algebraic relations (8.5) tell us that a k and a k(N/2−1) belong to the same conjugacy class and also that b and a m(N/2−2) b belong to the same conjugacy class. The group QD2N has the following (3 + N/2) conjugacy classes: C1 : C2[k] C1 :

:

CN/2 :  CN/2

:

{e},  k k(N/2−1)  , a ,a  N/2  , a   b, a 2 b, . . . , a N−2 b ,   ab, a 3 b, . . . , a N−1 b ,

h = 1, h = N/gcd(N, k), h = 2,

(8.7)

h = 2, h = 4,

where [k] = k or k(N/2 − 1) mod N with k = 1, . . . , N − 1 except N/2. We have also shown the order h of each element in the given conjugacy class. This implies that there are (3 + N/2) irreducible representations. Furthermore, the orthogonality relation (2.18) requires 

2  χα (C1 ) = mn n2 = m1 + 4m2 + 9m3 + · · · = 2N, (8.8) α

n

for the mi , which also satisfy m1 + m2 + m3 + · · · = 3 + N/2. The solution is (m1 , m2 ) = (4, N/2 − 1), so there are four singlets and (N/2 − 1) doublets.

8.1.2 Characters and Representations We now turn to the characters and representations. The group QD2N has four singlets and (N/2 − 1) doublets. We denote the four singlets by 1ss  , with s, s  = ±. The characters of a and b are obtained as χ1±s  (a) = ±1 for any s  and χ1s± (b) = ±1 for any s. Now consider the doublets. The two generators a and b are represented, e.g., by     k 0 ρ 0 1 , b= , (8.9) a= 1 0 0 ρ k(N/2−1)

8.1 Generic Aspects

71

Table 8.1 Characters of QD2N representations, where ρ = e2π i/N . Note that ρ mk + ρ mk(N/2−1) = 2 cos(2πmk/N) when mk is even, and ρ mk + ρ mk(N/2−1) = 2i sin(2πmk/N) when mk is odd h

χ1++

χ1−+

χ1−−

χ1+−

χ2[k]

C1

1

1

1

1

1

2

C2[m]

N/gcd(N, m)

1

(−1)m

(−1)m

1

ρ mk + ρ mk(N/2−1)

C1

2

1

(−1)N/2

CN/2

2

1

1

−1

−1

0

 CN/2

4

1

−1

1

−1

0

=1

(−1)N/2

=1

1

−2

on the doublet 2[k] . We also denote the vector 2[k] by  2[k] =



xk x(N/2−1)k

(8.10)

,

where each of the up and down components xk and x(N/2−1)k , respectively, has definite ZN charge. The characters are shown in Table 8.1.

8.1.3 Tensor Products We now discuss the tensor products of representations of the QD2N group. Let us start with 2k × 2k , i.e., 



xk x(N/2−1)k

 ⊗

yk  y(N/2−1)k 

2k

 (8.11)

, 2k 

where k, k  = 1, . . . , N − 1 except N/2. Hence, the tensor products of irreducible representations of QD2N are generally given by 

xk x(N/2−1)k  =



 ⊗ 2[k]



yk  y(N/2−1)k 

xk yk  x(N/2−1)k y(N/2−1)k 



2[k  ]

2[k+k  ]



 x y ⊕ k (N/2−1)k xN/2−1 kyk 

 , (8.12) 2[k+(N/2−1)k  ]

for k + k  , k + (N/2 − 1)k  = 0, N/2. In certain cases, the above representation becomes reducible. For example, if k + k  = 0 mod N , the doublet 2[k+k  ] can be

72

8

QD2N

reduced according to   xk yk  x(N/2−1)k y(N/2−1)k  2

[k+k  ]

= (xk yk + x(N/2−1)k y(N/2−1)k  )1++ ⊕ (xk yk  − x(N/2−1)k y(N/2−1)k  )1+− . (8.13) Similarly, when k + k  = N/2 mod N , the doublet 2[k+k  ] can be reduced according to   xk yk  x(N/2−1)k y(N/2−1)k  2  [k+k ]

= (xk yk + x(N/2−1)k y(N/2−1)k  )1−+ ⊕ (xk yk  − x(N/2−1)k y(N/2−1)k  )1−− . (8.14) The tensor products between singlets and doublets are:     xk wxk = , (w)1++ ⊗ x(N/2−1)k 2 wx(N/2−1)k 2 [k] [k]     xk wxk (w)1−− ⊗ = , x(N/2−1)k 2 −wx(N/2−1)k 2 [k] [k+N/2]     xk wxk (w)1+− ⊗ = , x(N/2−1)k 2 −wx(N/2−1)k 2 [k] [k]     xk wxk = . (w)1−+ ⊗ x(N/2−1)k 2 wx(N/2−1)k 2 [k]

(8.15) (8.16) (8.17) (8.18)

[k+N/2]

Finally, the tensor products among singlets are 1s1 s2 ⊗ 1s1 s2 = 1s1 s2 ,

(8.19)

where si = si si (i = 1, 2).

8.2 QD16 We discuss the simple example N  = 4 and N = 8, i.e., QD16 . This group is generated by the 2π/8 rotation a and the reflection b. These generators satisfy a 8 = 1,

b2 = 1,

bab = a 3 .

(8.20)

The group QD16 contains the 16 elements a m bk with m = 0, . . . , 7 and k = 0, 1. They are classified into the following seven conjugacy classes:

8.2 QD16

73

Table 8.2 Characters of QD16 representations

h

χ1++

C1

1

1

C2[1]

8

1

4

1

8

1

C1

2

1

C4

2

C4

2

C2[2] C2[5]

χ1−+

χ1−−

χ1+−

χ 21

χ2 2

1

1

1

−1

1

2 √ 2i

2

−1 1

1

1

0

−1

−1

1

√ − 2i

1

1

1

−2

2

−2

1

1

−1

−1

0

0

0

1

−1

1

−1

0

0

0

C1 :

h = 1,

C2[1] C2[2] C2[5] C1 :

{e},   : a, a 3 ,   : a2, a6 ,   : a5, a7 ,  4 a ,   C4 : b, a 2 b, . . . , a 6 b ,   C4 : ab, a 3 b, . . . , a 7 b ,

h = 8,

χ2 5 2

√ − 2i

0 −2

0 √

0

2i

h = 4, h = 8,

(8.21)

h = 2, h = 2, h = 4.

QD16 has four singlets 1±± , and three doublets 21 , 22 , and 23 . Their characters are shown in Table 8.2. We define the three doublets       x1 x2 x5 21 = , 22 = , 25 = . (8.22) x3 x6 x7 Then the tensor products are       y x1 y 1 x1 ⊗ 1 = ⊕ (x1 y3 ± x3 y1 )1−± , x3 2 y3 2 x3 y3 2 1 1 2         x1 y x3 y 2 x y ⊗ 2 = ⊕ 3 6 , x3 2 y6 2 x1 y6 2 x1 y2 2 1 2 5 1       x1 y5 x3 y 7 ⊗ = ⊕ (x1 y7 ± x3 y5 )1+± , x3 2 y7 2 x1 y5 2 1 5 2     x2 y ⊗ 2 = (x2 y2 ± x6 y6 )1−± ⊕ (x2 y6 ± x6 y2 )1+± , x6 2 y6 2 2 2         y5 x6 y 7 x2 y7 x2 ⊗ = ⊕ , x6 2 y7 2 x2 y5 2 x6 y5 2 2 5 5 1       x5 y x5 y 5 ⊗ 5 = ⊕ (x5 y7 ± x7 y5 )1−± , x7 2 y7 2 x7 y7 2 5 5 2         y wy1 y wy6 (w)1−± ⊗ 1 = , (w)1−± ⊗ 2 = y3 2 ±wy3 2 y6 2 ±wy2 2 1

5

2

2

(8.23) (8.24) (8.25) (8.26) (8.27) (8.28) , (8.29)

74

8

 (w)1−± ⊗

y5 y7



 =

25

wy5 ±wy7

1+± ⊗ 1+± = 1++ ,

QD2N

 ,

(8.30)

21

1−± ⊗ 1+± = 1−+ ,

1−± ⊗ 1−± = 1++ , (8.31)

1+± ⊗ 1+∓ = 1+− ,

1−± ⊗ 1+∓ = 1−− ,

1−± ⊗ 1−∓ = 1+− . (8.32)

Chapter 9

Σ(2N 2 )

9.1 Generic Aspects In this chapter, we investigate the discrete group Σ(2N 2 ), which is isomorphic to  )  Z . Let us denote the generators of Z and Z  by a and a  , respec(ZN × ZN 2 N N tively, and the Z2 generator by b. These generators satisfy a N = a  = b2 = e, N

aa  = a  a,

bab = a  .

(9.1)

Therefore, all elements of Σ(2N 2 ) are given by g = bk a m a n , for k = 0, 1 and m, n = 0, 1, . . . , N − 1. Since these generators, a, a  , and b, can be represented, e.g., by       1 0 ρ 0 0 1 a= , a = , b= , 0 ρ 0 1 1 0

(9.2)

(9.3)

where ρ = e2πi/N , all elements of Σ(2N 2 ) can be expressed by the 2 × 2 matrices  m    ρ 0 0 ρm , . (9.4) 0 ρn ρn 0

9.1.1 Conjugacy Classes The conjugacy classes of Σ(2N 2 ) are easily found using the algebraic relations     b a l a m b−1 = a m a l , b ba l a m b−1 = ba m a l , (9.5)     a k ba l a m a −k = ba l−k a m+k , a k ba l a m a −k = ba l+k a m−k , H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_9, © Springer-Verlag Berlin Heidelberg 2012

75

9 Σ(2N 2 )

76

which are given by (9.3). Hence, we find that the group Σ(2N 2 ) has the following conjugacy classes: C1 : (1) C1

:

.. . (k)

C1 : .. . C1(N−1) : (k) CN :

C2(l,m) :

{e}, h = 1,   h = N, aa , .. .. . .  k k  h = N/gcd(N, k), a a , (9.6) .. .. . .  N−1 N−1  , h = N/gcd(N, N − 1), a a   k k−1  k k+1 N−1 , h = 2N/gcd(N, k), ba , ba a , . . . , ba , . . . , ba a  l m l m  h = N/gcd(N, l, m), a a ,a a , (l,m)

where l > m for l, m = 0, . . . , N − 1. The number of conjugacy classes C2 given by N(N − 1)/2 and the total number of conjugacy classes of Σ(2N 2 ) is   N (N − 1)/2 + N + N = N 2 + 3N /2.

is

9.1.2 Characters and Representations The orthogonality relations (2.18) and (2.19) for Σ(2N 2 ) give   m1 + m2 + · · · = N 2 + 3N /2. m1 + 22 m2 + · · · = 2N 2 , The solution is

(9.7)

  (m1 , m2 ) = 2N, N (N − 1)/2 ,

so there are 2N singlets and N (N − 1)/2 doublets. Let us first discuss the singlets. Since a and a  belong to the same conjugacy class (1,0) C2 , the characters χα (g) for the singlets should satisfy χα (a) = χα (a  ). Because b2 = e and a N = e, possible values of χα (g) for the singlets are χα (a) = ρ n and χα (b) = ±1. Then we have a total of 2N combinations, which correspond to the 2N singlets 1±n for n = 0, 1, . . . , N − 1. These characters are summarized in Table 9.1. Now consider the doublet representations. In (9.3), the generators are represented in the doublet representation. Similarly, (2 × 2) matrix representations for generic doublets 2p,q are obtained by replacing a → a p a q

and a  → a q a p .

That is, for doublets 2p,q , the generators a, a  , and b are  p      q ρ 0 0 0 1 ρ  , a = , b= . a= 0 ρp 0 ρq 1 0

(9.8)

(9.9)

9.1 Generic Aspects

77

Table 9.1 Characters of Σ(2N 2 ) representations h

χ1+n

χ1−n

C1

1

1

1

2

C1(1) .. .

N

ρ 2n

ρ 2n

2ρ p+q

C1(N −1)

N/gcd(N, N − 1)

ρ 2n(N −1)

ρ 2n(N −1)

2ρ (N −1)(p+q)

(k) CN

2N/gcd(N, k)

ρ kn

−ρ kn

0

N/gcd(N, l, m)

ρ (l+m)n

ρ (l+m)n

ρ lq+mp + ρ lp+mq

(l,m) C2

Let us denote the doublet 2p,q by

 2p,q =

xq xp

χ2p,q

 (9.10)

,

where we take p > q and q = 0, 1, . . . , N − 2. Then, each of the up and down  charge. That is, x components xq and xp , respectively, has a definite ZN × ZN q  and xp have (q, 0) and (0, p) ZN × ZN charges, respectively. The characters for the doublets are also summarized in Table 9.1.

9.1.3 Tensor Products We now consider the tensor products of doublets 2p,q in Σ(2N 2 ). Taking into ac charges, their tensor products are given by count the ZN × ZN         yq  xq yq  xp yq  xq ⊗ = ⊕ , (9.11) xp 2 yp  2   xp yp 2   xq yp  2   q,p

q ,p

q+q ,p+p

q +p,q+p

for q + q  = p + p  mod(N ) and q + p  = p + q  mod(N ),     xq yq  ⊗ = (xq yq  + xp yp )1+,q+q  ⊕ (xq yq  − xp yp )1−,q+q  xp 2 yp 2   q,p q ,p   xp yq  ⊕ , (9.12) xq yp  2   q +p,q+p

for q + q  = p + p  mod(N ) and q + p  = p + q  mod(N ),     xq yq  ⊗ = (xp yq  + xq yp )1+,q+p ⊕ (xp yq  − xq yp )1−,q+p xp 2 yp 2   q,p q ,p   xq yq  ⊕ , (9.13) xp yp 2   q+q ,p+p

9 Σ(2N 2 )

78

for q + q  = p + p  mod(N ) and q + p  = p + q  mod(N ), and     xq yq  ⊗ = (xq yq  + xp yp )1+,q+q  ⊕ (xq yq  − xp yp )1−,q+q  xp 2 yp  2   q,p

q ,p

⊕ (xp yq  + xq yp )1+,q+p ⊕ (xp yq  − xq yp )1−,q+p , (9.14) for q + q  = p + p  mod(N ) and q + p  = p + q  mod(N ). Furthermore, we obtain the tensor products between singlets and doublets as     xq yxq (y)1s,n ⊗ = . (9.15) xp 2 yxp 2 q,p

q+n,p+n

These tensor products are independent of s = ±. The tensor products of singlets are simply given by 1sn ⊗ 1s  n = 1ss  ,n+n .

(9.16)

9.2 Σ(18) In this and the following two sections, we present some simple examples of Σ(2N 2 ). The simplest group Σ(2N 2 ) is Σ(2), which is nothing but the Abelian group Z2 . The next is the group Σ(8), which is isomorphic to D4 . Consequently, the simplest non-trivial example is Σ(18). In Σ(18), there are eighteen elements bk a m a n for k = 0, 1, and m, n = 0, 1, 2, where a, a  , and b satisfy b2 = e, a 3 = a 3 = e, aa  = a  a, and bab = a  . These elements are classified into nine conjugacy classes: C1 : (1) C1 : (2) C1 : (0) C3 : (1) C3 : (2) C3 : (1,0) : C2 (2,0) C2 : (2,1) C2 :

{e},   aa ,  2 2  a a ,   b, ba 2 a, ba  a 2 ,    ba , ba, ba 2 a 2 ,   2 ba , ba  a, ba 2 ,   a, a  ,  2 2  a ,a ,  2   a a , aa 2 ,

h = 1, h = 3, h = 3, h = 2, h = 6, h = 6, h = 3, h = 3, h = 3,

where h is the order of each element in the given conjugacy class.

(9.17)

9.2 Σ(18)

79

Table 9.2 Characters of Σ(18) representations

h χ1+0 χ1+1 χ1+2 χ1−0 χ1−1 χ1−2 χ21,0 χ22,0 χ22,1 C1

1 1

1

1

1

1

1

2

2

2

C1(1)

3 1

ρ2

ρ

1

ρ2

ρ

2ρ

2ρ 2

2

3 1

ρ

ρ2

ρ

ρ2

2ρ 2

2ρ

2

2 1

1

1

−1

−1

−1

0

0

0

6 1

ρ

ρ2

−1

−ρ

−ρ 2 0

0

0

6 1

ρ2

ρ

−1

−ρ 2

−ρ

0

0

0

3 1

ρ

ρ2

1

ρ

ρ2

−ρ 2 −ρ −1

3 1

ρ2

ρ

1

ρ2

ρ

−ρ

−ρ 2 −1

3 1

1

1

1

1

1

−1

−1 −1

C1(2) C3(0) C3(1) (2) C3 C2(1,0) (2,0) C1 C1(3,0)

1

Σ(18) has six singlets 1±,n with n = 0, 1, 2, and three doublets 2p,q with (p, q) = (1, 0), (2, 0), (2, 1). The characters are shown in Table 9.2. The tensor products between doublets are as follows:       x2 y2 x1 y 1 ⊗ = (x1 y2 + x2 y1 )1+,0 ⊕ (x1 y2 − x2 y1 )1−,0 ⊕ , x1 2 y1 2 x2 y2 2 2,1

    

x2 x0 x1 x0 x2 x1 x2 x1 x2 x0

2,1



 ⊗

22,0



 ⊗ 21,0

 22,1

y1 y0



y ⊗ 2 y0 

 ⊗ 22,1



 ⊗

22,0

y2 y0

y1 y0 y1 y0

2,1



 22,0

= (x0 y2 + x2 y0 )1+,2 ⊕ (x0 y2 − x2 y0 )1−,2 ⊕ 

 21,0

= (x0 y1 + x1 y0 )1+,1 ⊕ (x0 y1 − x1 y0 )1−,1 ⊕

x2 y 2 x0 y0 x1 y 1 x0 y0



 22,0

x y = (x2 y2 + x1 y0 )1+,1 ⊕ (x2 y2 − x1 y0 )1−,1 ⊕ 2 0 x1 y2 

 21,0

= (x1 y1 + x2 y0 )1+,2 ⊕ (x1 y1 − x2 y0 )1−,2 ⊕



 21,0

= (x2 y1 + x0 y0 )1+,0 ⊕ (x2 y1 − x0 y0 )1−,0 ⊕

x1 y 0 x2 y1 x2 y 0 x0 y1

 , 21,0

 , 22,0

 , 22,0

 , 21,0

 . 22,1

(9.18) The tensor products between singlets are: 1±,0 ⊗ 1±,0 = 1+,0 ,

1±,1 ⊗ 1±,1 = 1+,2 ,

1±,2 ⊗ 1±,2 = 1+,1 ,

1±,1 ⊗ 1±,0 = 1+,1 ,

1±,2 ⊗ 1±,0 = 1+,2 ,

1±,2 ⊗ 1±,1 = 1+,0 ,

1±,0 ⊗ 1∓,0 = 1−,0 ,

1±,1 ⊗ 1∓,1 = 1−,2 ,

1±,2 ⊗ 1∓,2 = 1−,1 ,

1±,1 ⊗ 1∓,0 = 1−,1 ,

1±,2 ⊗ 1±,0 = 1−,2 ,

1±,2 ⊗ 1∓,1 = 1−,0 .

(9.19)

9 Σ(2N 2 )

80

And finally, the tensor products between singlets and doublets are:         x yx2 x yx1 = , (y)1±,1 ⊗ 2 = , (y)1±,0 ⊗ 2 x1 2 yx1 2 x1 2 yx2 2 2,1 2,1 2,1 2,0         x2 yx2 x2 yx2 = , (y)1±,0 ⊗ = , (y)1±,2 ⊗ x1 2 yx1 2 x0 2 yx0 2 2,1 1,0 2,0 2,0         x2 yx0 x2 yx0 (y)1±,1 ⊗ = , (y)1±,2 ⊗ = , x0 2 yx2 2 x0 2 yx2 2 (9.20) 2,0 1,0 2,0 2,1         x yx1 x yx1 = , (y)1±,1 ⊗ 1 = , (y)1±,0 ⊗ 1 x0 2 yx0 2 x0 2 yx0 2 1,0 1,0 1,0 2,1     x1 yx0 (y)1±,2 ⊗ = . x0 2 yx1 2 1,0

2,0

9.3 Σ(32) The next example is the group Σ(32), which has thirty-two elements, bk a m a n for k = 0, 1, and m, n = 0, 1, 2, 3, where a, a  , and b satisfy b2 = e, a 4 = a 4 = e, aa  = a  a, and bab = a  . These elements are classified into fourteen conjugacy classes: C1 : {e}, h = 1,   (1) C1 : h = 4, aa ,  2 2  (2) C1 : h = 2, a a ,  3 3  (3) C1 : h = 4, a a ,   (0) b, ba  a 3 , ba 2 a 2 , ba 3 a , h = 2, C4 :   C4(1) : ba  , ba, ba 2 a 3 , ba 3 a 2 , h = 8,   2 (2) C4 : ba , ba  a, ba 2 , ba 3 a 3 , h = 4, (9.21)   3 (3) C4 : ba , ba 2 a, ba  a 2 , ba 3 , h = 8,   (1,0) C2 h = 4, : a, a  ,  2 2  (2,0) h = 2, : a ,a , C2   2  (2,1) 2 C2 h = 4, : a a , aa ,  3 3  (3,0) C2 h = 4, : a ,a ,   3  (3,1) 3 C2 h = 4, : a a , aa ,  3 2 2 3  (3,2) h = 4, : a a ,a a , C2 where h is the order of each element in the given conjugacy class. Σ(32) has eight singlets 1±,n with n = 0, 1, 2, 3, and six doublets 2p,q with (p, q) = (1, 0), (2, 0), (3, 0), (2, 1), (3, 1), (3, 2). The characters are shown in Table 9.3.

9.3 Σ(32)

81

Table 9.3 Characters of Σ(32) representations h χ1+0 χ1+1 χ1+2 χ1+3 χ1−0 χ1−1 χ1−2 χ1−3 χ21,0

χ22,0 χ22,1

χ23,0

χ23,1 χ23,2

C1

1 1

1

1

1

1

1

1

1

2

2

2

2

2

2

C1(1)

4 1

−1

1

−1

1

−1

1

−1

2i

−2

−2i

−2i

2

2i

C1

(2)

2 1

1

1

1

1

1

1

1

−2

2

−2

−2

2

−2

C1(3)

4 1

−1

1

−1

1

−1

1

−1

−2i

−2

2i

2i

2

−2i

C4

2 1

1

1

1

−1

−1

−1

0

0

0

0

0

0

C4(1) 8 1

−1

i

−1

−i

−1

−i

1

i

0

0

0

0

0

0

4 1

−1

1

−1

−1

1

−1

1

0

0

0

0

0

0

8 1

−i

−1

i

−1

i

1

−i

0

0

0

0

0

0

4 1

i

−1

−i

1

i

−1

−i

1+i

0

−1 + i 1 − i

0

−1 − i

2 1

−1

1

−1

1

−1

1

−1

0

2

0

0

4 1

−i

−1

i

1

−i

−1

i

−1 + i 0

1+i

−1 − i 0

4 1

−i

−1

i

1

−i

−1

i

1−i

−1 − i 1 + i

4 1

1

1

1

1

1

1

1

4 1

i

−1

−i

1

i

−1

−i

(0)

C4(2) C4(3) C2(1,0) C2(2,0) C2(2,1) C2(3,0) C2(3,1) (3,2) C2

0

0 −2

−1 − i 0

−2

0 1−i −1 + i

0 −2

0

0

1−i

−1 + i 0

0 1+i

The tensor products between doublets are as follows:       x3 y3 x3 y3 ⊗ = (x2 y3 + x3 y2 )1+,1 ⊕ (x2 y3 − x3 y2 )1−,1 ⊕ , x2 2 y2 2 x2 y2 2 3,2











x3 x1

x3 x0 x2 x1 x2 x0

x1 x0

3,2



 ⊗ 23,1



 ⊗ 23,0



y3 y1

y3 y0



22,1

y ⊗ 2 y1



 ⊗ 22,0



 ⊗ 21,0

y2 y0

y1 y0

2,0

(9.22)

 23,1

 23,0

 22,1

= (x1 y3 + x3 y1 )1+,0 ⊕ (x1 y3 − x3 y1 )1−,0 ⊕ (x3 y3 + x1 y1 )1+,2 ⊕ (x3 y3 − x1 y1 )1−,2 , (9.23)   x y = (x0 y3 + x3 y0 )1+,3 ⊕ (x0 y3 − x3 y0 )1−,3 ⊕ 3 3 , x0 y0 2 2,0



x y = (x1 y2 + x2 y1 )1+,3 ⊕ (x1 y2 − x2 y1 )1−,3 ⊕ 1 1 x2 y2

 21,0

, 22,0

(9.25)

 22,0



(9.24)

= (x0 y2 + x2 y0 )1+,2 ⊕ (x0 y2 − x2 y0 )1−,2 ⊕ (x2 y2 + x0 y0 )1+,0 ⊕ (x2 y2 − x0 y0 )1−,0 , (9.26)   x y = (x0 y1 + x1 y0 )1+,1 ⊕ (x0 y1 − x1 y0 )1−,1 ⊕ 1 1 , x0 y0 2 2,0

(9.27)

9 Σ(2N 2 )

82

 



 



x3 x2 x3 x2

x3 x2

x3 x2 x3 x2

x3 x1



 ⊗



23,2

 ⊗

23,2



 ⊗ 23,2



 ⊗



23,2

 ⊗

23,2



 ⊗

y3 y0

y2 y1

y2 y0 y1 y0

y3 y0



 =

23,1

x2 y 3 x3 y1

 21,0



23,0

 ⊕

x2 y 1 x3 y3

 (9.28)

, 23,2



= (x3 y3 + x2 y0 )1+,2 ⊕ (x3 y3 − x2 y0 )1−,2 ⊕ 

 22,1

= (x2 y2 + x3 y1 )1+,0 ⊕ (x2 y2 − x3 y1 )1−,0 ⊕



 =

22,0

x3 y 0 x2 y2

 ⊕ 23,1



21,0



x2 y 0 x3 y2

 =

23,0

x3 y 0 x1 y3



 ⊕ 23,0

x3 y 3 x1 y0

x2 y1 x3 y2

 , 23,1



x3 x1



3,1

 

 



x3 x1

, (9.30) (9.31)

, 22,1



x2 y0 x3 y1

 , 22,0

(9.32)



(9.33)

, 22,1

x3 x0

x3 x0 x3 x0

x2 x1

 ⊗ 23,1

 ⊗

23,0



 ⊗ 

23,0

23,0

y1 y0 y2 y1

y2 y0



y ⊗ 1 y0 

 ⊗ 22,1

(9.34)

2,0

 

(9.29)

23,1



= (x2 y1 + x3 y0 )1+,3 ⊕ (x2 y1 − x3 y0 )1−,3 ⊕



x3 y0 x2 y3

     y2 x1 y 2 x1 y 1 ⊗ = ⊕ , y1 2 x3 y1 2 x3 y2 2 23,1 2,1 3,0 2,1     y x3 ⊗ 2 = (x1 y2 + x3 y0 )1+,3 ⊕ (x1 y2 − x3 y0 )1−,3 x1 2 y0 2 



23,1

y3 y1

y2 y0

 21,0



22,1

⊕ (x3 y2 + x1 y0 )1+,1 ⊕ (x3 y2 − x1 y0 )1−,1 , (9.35)     x3 y 0 x y = ⊕ 1 0 , (9.36) x1 y1 2 x3 y1 2 3,2 1,0   x y = (x3 y2 + x0 y1 )1+,1 ⊕ (x3 y2 − x0 y1 )1−,1 ⊕ 0 2 , x3 y1 2 2,0



 = 22,0



21,0

x3 y 0 x0 y2



 ⊕ 23,2

x3 y 2 x0 y0

(9.37)

 

x y = (x3 y1 + x0 y0 )1+,0 ⊕ (x3 y1 − x0 y0 )1−,0 ⊕ 3 0 x0 y1 

 22,0

(9.38)

, 21,0

= (x2 y2 + x1 y0 )1+,1 ⊕ (x2 y2 − x1 y0 )1−,1 ⊕

x1 y2 x2 y0

 , 23,1



(9.39) , 23,2

(9.40)

9.3 Σ(32)





x2 x1

x2 x0

83



 ⊗ 22,1



 ⊗ 22,0

y1 y0

y1 y0





21,0

= (x1 y1 + x2 y0 )1+,2 ⊕ (x1 y1 − x2 y0 )1−,2 ⊕ 

 = 21,0

x2 y 1 x0 y0



 ⊕ 22,0

x2 y 0 x0 y1

x2 y1 x1 y0

 , 23,1

(9.41)



(9.42)

. 22,1

The tensor products between singlets are: 1±,0 ⊗ 1±,0 = 1+,0 ,

1±,1 ⊗ 1±,1 = 1+,2 ,

1±,2 ⊗ 1±,2 = 1+,0 ,

1±,3 ⊗ 1±,3 = 1+,2 ,

1±,3 ⊗ 1±,2 = 1+,1 ,

1±,3 ⊗ 1±,1 = 1+,0 ,

1±,3 ⊗ 1±,0 = 1+,3 ,

1±,2 ⊗ 1±,1 = 1+,3 ,

1±,2 ⊗ 1±,0 = 1+,2 ,

1±,1 ⊗ 1±,0 = 1+,1 ,

1∓,0 ⊗ 1±,0 = 1−,0 ,

1∓,1 ⊗ 1±,1 = 1−,2 , (9.43)

1∓,2 ⊗ 1±,2 = 1−,0 ,

1∓,3 ⊗ 1±,3 = 1−,2 ,

1∓,3 ⊗ 1±,2 = 1−,1 ,

1∓,3 ⊗ 1±,1 = 1−,0 ,

1∓,3 ⊗ 1±,0 = 1−,3 ,

1∓,2 ⊗ 1±,1 = 1−,3 ,

1∓,2 ⊗ 1±,0 = 1−,2 ,

1∓,1 ⊗ 1±,0 = 1−,1 .

Finally, the tensor products between singlets and doublets are:  x3 (y)1±,0 ⊗ x2 2 3,2   x (y)1±,2 ⊗ 3 x2 2 3,2   x (y)1±,0 ⊗ 3 x1 2 3,1   x (y)1±,2 ⊗ 3 x1 2 3,1   x (y)1±,0 ⊗ 3 x0 2 3,0   x (y)1±,2 ⊗ 3 x0 2 3,0   x (y)1±,0 ⊗ 2 x1 2 2,1   x (y)1±,2 ⊗ 2 x1 2 

2,1

 =  =  =  =  =  =  =  =

yx3 yx2 yx3 yx2 yx3 yx1 yx1 yx3 yx3 yx0 yx0 yx3 yx2 yx1 yx1 yx2

 , 

23,2

, 

21,0

, 

23,1

, 

23,1

, 

23,0

, 

22,1

, 

 x3 (y)1±,1 ⊗ x2 2 3,2   x (y)1±,3 ⊗ 3 x2 2 3,2   x (y)1±,1 ⊗ 3 x1 2 3,1   x (y)1±,3 ⊗ 3 x1 2 3,1   x (y)1±,1 ⊗ 3 x0 2 3,0   x (y)1±,3 ⊗ 3 x0 2 3,0   x (y)1±,1 ⊗ 2 x1 2 2,1   x (y)1±,3 ⊗ 2 x1 2 

22,1

, 23,0

2,1

 =  =  =  =  =  =  =  =

yx2 yx3 yx3 yx2 yx1 yx3 yx3 yx1 yx0 yx3 yx0 yx3 yx2 yx1 yx2 yx1

 , 

23,0

, 

22,1

, 

22,0

, 

22,0

, 

21,0

, 

23,2

, 

23,2

, 21,0

84

9

 (y)1±,0 ⊗  (y)1±,2 ⊗

x2 x0 x2 x0



 =

22,0



 =

22,0

yx2 yx0 yx0 yx2



 ,



22,0

 ,

22,0

   x1 yx1 = , (y)1±,0 ⊗ x0 2 yx0 2 1,0 1,0     x yx1 = , (y)1±,2 ⊗ 1 x0 2 yx0 2 

1,0

3,2

(y)1±,1 ⊗ (y)1±,3 ⊗

x2 x0 x2 x0



 =



22,0

 =

22,0

yx2 yx0 yx0 yx2

Σ(2N 2 )

 , 

23,1

, 23,1

   x1 yx1 (y)1±,1 ⊗ = , x0 2 yx0 2 1,0 2,1     x yx0 (y)1±,3 ⊗ 1 = . x0 2 yx1 2 

1,0

3,0

(9.44)

9.4 Σ(50) 

This group has fifty elements, bk a m a n for k = 0, 1, and m, n = 0, 1, 2, 3, 4, where a, a  , and b satisfy the same conditions as (9.1) for the case N = 5. These elements are classified into twenty conjugacy classes: C1 : (1) C1 : (2) C1 : (3) C1 : (4) C1 : C5(0) : C5(1) : (2) C5 : (3) C5 : (4) C5 : (1,0) : C2 (2,0) C2 : (2,1) C2 : (3,0) C2 : (3,1) C2 : (3,2) C2 : (4,0) C2 : (4,1) : C2 (4,2) C2 : (4,3) C2 :

{e},   aa ,  2 2  a a ,  3 3  a a ,  3 3  a a ,   b, ba 2 a 3 , ba 3 a 2 , ba 4 a, ba  a 4 ,    ba , ba, ba 3 a 3 , ba 4 a 2 , ba 2 a 4 ,   2 ba , ba  a, ba 2 , ba 4 a 3 , ba 3 a 4 ,   3 ba , ba 2 a, ba  a 2 , ba 3 , ba 4 a 4 ,   4 ba , ba 2 a 2 , ba  a 3 , ba 3 a, ba 4 ,   a, a  ,  2 2  a ,a ,   2  a a , aa 2 ,  3 3  a ,a ,   3  a a , aa 3 ,  3 2 2 3  a a ,a a ,  4 4  a ,a ,   4  a a , aa 4 ,  4 2 2 4  a a ,a a ,  4 3 3 4  a a ,a a ,

h = 1, h = 5, h = 5, h = 5, h = 5, h = 2, h = 10, h = 10, h = 10, h = 10, h = 5, h = 5, h = 5, h = 5, h = 5, h = 5, h = 5, h = 5, h = 5, h = 5,

where h is the order of each element in the given conjugacy class.

(9.45)

9.4

85

Σ(50)

Table 9.4 Characters of Σ(50) representations, where ρ = e2iπ/5 C1 C1(1) C1(2) (3) C1 (4) C1 C5(0) C5(1) C5(2) C5(3) (4) C5 (1,0) C2 C2(2,0) C2(2,1) C2(3,0) C2(3,1) (3,2) C2 C2(4,0) C2(4,1) C2(4,2) C2(4,3)

h

χ1±0

χ1±1

χ1±2

χ1±3

χ1±4

1 5 5 5 5 2 10 10 10 10 5 5 5 5 5 5 5 5 5 5

1 1 1 1 1 ±1 ±1 ±1 ±1 ±1 1 1 1 1 1 1 1 1 1 1

1 ρ2 ρ4 ρ ρ3 ±1 ±ρ ±ρ 2 ±ρ 3 ±ρ 4 ρ ρ2 ρ3 ρ3 ρ4 1 ρ4 1 ρ ρ2

1 ρ4 ρ3 ρ2 ρ ±1 ±ρ 2 ±ρ 4 ±ρ ±ρ 3 ρ2 ρ4 ρ ρ ρ3 1 ρ3 1 ρ2 ρ4

1 ρ ρ2 ρ3 ρ4 ±1 ±ρ 3 ±ρ ±ρ 4 ±ρ 2 ρ3 ρ ρ4 ρ4 ρ2 1 ρ2 1 ρ3 ρ

1 ρ3 ρ ρ4 ρ2 ±1 ±ρ 4 ±ρ 3 ±ρ 2 ±ρ ρ3 ρ3 ρ2 ρ2 ρ 1 ρ 1 ρ4 ρ3

Table 9.5 Characters of Σ(50) representations, where ρ = e2iπ/5 h χ21,0 C1 1 C1(1) 5 C1(2) 5 C1(3) 5 (4) C1 5 (0) C5 2 C5(1−4) 10 C2(1,0) 5 C2(2,0) 5 (2,1) C2 5 (3,0) 5 C2 C2(3,1) 5 C2(3,2) 5 C2(4,0) 5 C2(4,1) 5 (4,2) C2 5 (4,3) C2 5

χ22,0

χ22,1

χ23,0

χ23,1

χ23,2

χ24,0

χ24,1

χ24,2

χ24,3

2 2 2 2 2 2 2 2 2 2 2ρ 2ρ 2 2ρ 3 2ρ 3 2ρ 4 2 2ρ 4 2 2ρ 2ρ 2 2 4 3 3 2 2ρ 2ρ 2ρ 2ρ 2ρ 2 2ρ 2 2ρ 2ρ 4 3 4 4 2 2 3 2ρ 2ρ 2ρ 2ρ 2ρ 2 2ρ 2 2ρ 2ρ 2ρ 4 2ρ 3 2ρ 2 2ρ 2 2ρ 2 2ρ 2 2ρ 4 2ρ 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 + ρ 1 + ρ2 ρ + ρ2 1 + ρ3 ρ + ρ3 ρ2 + ρ3 1 + ρ4 ρ + ρ4 ρ2 + ρ4 ρ3 + ρ4 1 + ρ2 1 + ρ4 ρ2 + ρ4 1 + ρ ρ + ρ2 ρ + ρ4 1 + ρ3 ρ2 + ρ3 ρ3 + ρ4 ρ + ρ3 ρ + ρ2 ρ2 + ρ4 1 + ρ4 ρ + ρ3 1 + ρ2 ρ2 + ρ3 ρ3 + ρ4 ρ + ρ4 1 + ρ3 1 + ρ 1 + ρ3 1 + ρ ρ + ρ3 1 + ρ4 ρ3 + ρ4 ρ + ρ4 1 + ρ2 ρ2 + ρ3 ρ + ρ2 ρ2 + ρ4 ρ + ρ3 ρ + ρ2 1 + ρ2 ρ3 + ρ4 1 + ρ ρ + ρ4 ρ2 + ρ4 ρ2 + ρ3 1 + ρ4 1 + ρ3 ρ2 + ρ3 ρ + ρ4 ρ2 + ρ3 ρ + ρ4 ρ + ρ4 ρ2 + ρ3 ρ2 + ρ3 ρ + ρ4 ρ + ρ4 ρ2 + ρ3 1 + ρ4 1 + ρ3 ρ3 + ρ4 1 + ρ2 ρ2 + ρ4 ρ2 + ρ3 1 + ρ ρ + ρ4 ρ + ρ3 ρ + ρ2 ρ + ρ4 ρ2 + ρ3 ρ + ρ4 ρ2 + ρ3 ρ2 + ρ3 ρ + ρ4 ρ + ρ4 ρ2 + ρ3 ρ2 + ρ3 ρ + ρ4 ρ2 + ρ4 ρ3 + ρ4 1 + ρ3 ρ + ρ2 1 + ρ4 1 + ρ ρ + ρ3 ρ2 + ρ3 1 + ρ 1 + ρ2 ρ3 + ρ4 ρ + ρ3 1 + ρ ρ2 + ρ4 1 + ρ3 ρ2 + ρ3 ρ + ρ2 ρ + ρ4 1 + ρ2 1 + ρ4

86

9

Σ(2N 2 )

The group Σ(50) has ten singlets 1±,n with n = 0, 1, 2, 3, 4, and ten doublets 2p,q with (p, q) = (1, 0), (2, 0), (3, 0), (4, 0), (2, 1), (3, 1), (4, 1), (3, 1), (3, 2), (4, 3). The characters are shown in Tables 9.4 and 9.5. We omit the explicit expressions of the tensor products since they can be obtained in the same way as for Σ(18) and Σ(32).

Chapter 10

Δ(3N 2 )

In this chapter, we investigate the discrete group Δ(3N 2 ), which is isomorphic to  )  Z (see also [1]). The generators of Z and Z  are denoted by a and (ZN × ZN 3 N N  a , respectively, and the Z3 generator is written b. These generators satisfy a N = a  N = b3 = e,  −1 bab−1 = a −1 a  ,

aa  = a  a,

(10.1)

ba  b−1 = a.

Therefore, all the elements of Δ(3N 2 ) can be written in the form g = bk a m a n , for k = 0, 1, 2, and m, n = 0, 1, 2, . . . , N − 1. Since the generators, a, a  , and b, are represented, e.g., by ⎛ ⎞ ⎛ ⎞ ⎛ −1 ρ 0 0 0 1 0 ρ 0 ⎠, b = ⎝0 0 1⎠, a =⎝0 1 a = ⎝ 0 1 0 0 0 0 ρ −1 0

(10.2)

0 ρ 0

⎞ 0 0 ⎠ , (10.3) 1

where ρ = e2πi/N , all elements of Δ(3N 2 ) can be written in the form ⎛ m ⎞ ⎛ ⎞ ⎛ ρ 0 0 0 ρm 0 0 0 ⎝ 0 pn ⎝ 0 ⎝ ρn 0 ⎠, 0 ρn ⎠ , 0 0 0 ρ −m−n ρ −m−n 0 0 0 ρ −m−n

⎞ ρm 0 ⎠, 0 (10.4)

for m, n = 0, 1, 2, . . . , N − 1.

10.1 Δ(3N 2 ) with N/3 = Integer The groups Δ(3N 2 ) have different features depending on whether N/3 is an integer or not. First, we study Δ(3N 2 ) when N/3 = integer. H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_10, © Springer-Verlag Berlin Heidelberg 2012

87

88

10

Δ(3N 2 )

10.1.1 Conjugacy Classes The conjugacy classes of Δ(3N 2 ) with N/3 = integer are found to be ba  a m b−1 = a −+m a − ,

b2 a  a m b−2 = a −m a −m .

(10.5)

Thus, these elements a  a m , a −+m a − , and a −m a −m , must belong to the same conjugacy class. They are independent elements of Δ(3N 2 ) unless N/3 integer and 3 =  + m = 0 (mod N ). As a result, the elements a  a m are classified into the following conjugacy classes:  (,m) C3 (10.6) = a  a m , a −+m a − , a −m a −m , for N/3 = integer. In the same way, we can obtain conjugacy classes containing ba  a m . Let us consider the conjugates of the simplest element b among ba  a m . We find a p a q (b)a −p a −q = ba −p−q a p−2q = ba −n+3q a n , where we have defined n ≡ p − 2q for convenience. We also get   b ba −n+3q a n b−1 = ba 2n−3q a n−3q ,   b2 ba −n+3q a n b−2 = ba −n a −2n+3q .

(10.7)

(10.8) (10.9)

The important thing to note here is that q appears only in the combination 3q. Thus, if N/3 = integer, the element b is conjugate to all elements ba  a m . That is, all (1) of them belong to the same conjugacy class CN 2 . Similarly, all elements b2 a  a m (2) belong to the same conjugacy class CN 2 when N/3 = integer. We summarize the conjugacy classes of Δ(3N 2 ) for N/3 = integer:

C1 : C3(,m) (1) CN 2 : (2) CN 2 :

:

{e},   m −+m − −m −m , a a ,a a ,a a   m ba a |, m = 0, 1, . . . , N − 1 ,  2  m b a a |, m = 0, 1, . . . , N − 1 ,

h = 1, h = N/ gcd(N, , m), h = 3,

(10.10)

h = 3.

(,m)

is (N 2 − 1)/3, whence the total number The number of conjugacy classes C3 of conjugacy classes is 3 + (N 2 − 1)/3. The relations (2.18) and (2.19) for Δ(3n2 ) with N/3 = integer give m1 + 22 m2 + 32 m3 + · · · = 3N 2 ,   m1 + m2 + m3 + · · · = 3 + N 2 − 1 /3,

(10.11) (10.12)

which implies that (m1 , m3 ) = (3, (N 2 − 1)/3). Therefore, we find three singlets and (N 2 − 1)/3 triplets.

10.1

Δ(3N 2 ) with N/3 = Integer

89

10.1.2 Characters and Representations There are 3 singlets in the group Δ(3N 2 ) with N/3 = integer. Since b3 = e is satisfied in this group, the characters of the three singlets have three possible values χ1k (b) = ωk with k = 0, 1, 2, which correspond to three singlets 1k . Because χ1k (b) = χ1k (ba) = χ1k (ba  ), we find that χ1k (a) = χ1k (a  ) = 1. These characters are given in Table 10.1. Now consider the triplet representations. As can be seen from (10.3), we have (3 × 3) matrices corresponding to one of the triplet representations. Therefore, (3 × 3) matrix representations for the generic triplets are found by replacing a → a  a m ,

a  → b2 ab−2 = a −m a −m .

(10.13)

However, we may note that the two types of replacement         a, a  → a −+m a − , a  a m , a, a  → a −m a −m , a −+m a − , (10.14) also lead to a representation equivalent to the above (10.13), because the three elements a  a m , a −+m a − , and a −m a −m belong to the same conjugacy class, viz., (,m) C3 . Thus, the generators of Δ(3N 2 ) with N/3 = integer are represented by ⎛ ⎞ ⎛  ⎛ −k− ⎞ ⎞ ρ ρ 0 0 0 0 0 1 0 b = ⎝0 0 1⎠, a = ⎝ 0 ρk a = ⎝ 0 0 ⎠, ρ 0 ⎠ , −k− 1 0 0 0 0 ρ 0 0 ρk (10.15) on the triplet 3[k][] , where the notation [k][] is defined by1 [k][] = (k, ), (−k − , k), or (, −k − ). We denote the vector 3[k][] by

⎞ x,−k− 3[k][] = ⎝ xk, ⎠ , x−k−,k

(10.16)

⎛

(10.17)

 charges, for k,  = 0, 1, . . . , N − 1, where k and  correspond to the ZN and ZN  respectively. When (k, ) = (0, 0), the matrices a and a are the identity matrices. Thus, we exclude the case with (k, ) = (0, 0). The characters are given in Table 10.1.

10.1.3 Tensor Products  First consider the tensor products of triplets. Taking into account the ZN × ZN charges, these are given by

1 [k][]

 corresponds to (k, ) in [1].

90

10

Δ(3N 2 )

Table 10.1 Characters of Δ(3N 2 ) for N/3 = integer h

χ 10

χ 11

χ 12

χ3[n][m]

C1

1

1

1

1

3

(k,) C3

N gcd(N, k, ) 3

1

1

1

ρ mk−n−m + ρ nk+m + ρ −nk−mk−n

1

ω

ω2

0

1

ω2

ω

0

(1)

CN 2

(2) CN 2

3

⎛

⎞ ⎞ ⎛ x,−k− y ,−k  − ⎝ xk, ⎠ ⊗ ⎝ yk , ⎠ x−k−,k 3 y−k  − ,k  3   [k][] [k ][ ] ⎞ ⎞ ⎛ ⎛ x,−k− y ,−k  − xk, y−k  − ,k  ⎠ xk, yk , =⎝ ⊕ ⎝ x−k−,k y ,−k  − ⎠ x−k−,k y−k  − ,k 3   x,−k− yk  , 3[−k−+ ][k−k  − ] [k+k ][+ ] ⎛ ⎞ x−k−,k yk , ⊕ ⎝ x,−k− y−k  − ,k ⎠ (10.18) xk, y ,−k  − 3    [−k − ][−k−+k ]

for −(k, ) = [k  ][ ], and ⎛ ⎛ ⎞ ⎞ y−,k+ x,−k− ⎝ xk, ⎠ ⊗ ⎝ y−k,− ⎠ x−k−,k 3 yk+,−k 3 [k][]

−[k][]

= (x,−k− y−,k+ + xk, y−k,− + x−k−,k yk+,−k )10   ⊕ x,−k− y−,k+ + ω2 xk, y−k,− + ωx−k−,k yk+,−k 1 1   2 ⊕ x,−k− y−,k+ + ωxk, y−k,− + ω x−k−,k yk+,−k 1 2 ⎛ ⎛ ⎞ ⎞ xk, yk+,−k x−k−,k y−k,− ⊕ ⎝ x−k−,k y−,k+ ⎠ ⊕ ⎝ x,−k− yk+,−k ⎠ x,−k− y−k,− 3 xk, y−,k+ 3 [−k−2][2k+]

.

[k+2][−2k−]

(10.19) The product of 3[k][] and 1r is ⎛ ⎞ ⎛ ⎞ x,−k− z x,−k− ⎝ xk, ⎠ ⊗ (z)1r = ⎝ ωr xk, z ⎠ . x−k−,k [k][] ω2r x−k−,k z [k][]

(10.20)

The tensor products of singlets 1k and 1k  are simply given by 1k ⊗ 1k = 1k+k  mod 3 .

(10.21)

10.2

Δ(3N 2 ) with N/3 Integer

91

10.2 Δ(3N 2 ) with N/3 Integer In this section, we study groups Δ(3N 2 ) with N/3 an integer.

10.2.1 Conjugacy Classes The algebraic relation (10.5) holds true when N/3 is an integer, as well as for the case N/3 = integer. Thus, the elements a  a m , a −+m a − , and a −m a −m , must belong to the same conjugacy class. When N/3 is an integer and 3 =  + m = 0 (mod N ), the above elements are the same, i.e., a  a − . As a result, the elements a  a m are classified into the following conjugacy classes:  C1(k) = a k a −k , (,m) C3

k=

N 2N , , 3 3

 = a  a m , a −+m a − , a −m a −m ,

(, m) =



 N 2N 2N N , , , , 3 3 3 3 (10.22)

for N/3 integer. Similarly, we find the conjugacy classes containing ba  a m . One can obtain the conjugates of b among ba  a m using (10.7), (10.8), and (10.9). When N/3 = integer, the element b is conjugate to all elements of the form ba  a m , as shown in the last section. However, the situation is different when N/3 integer. The elements conjugate to b do not include ba. The conjugates to ba are also obtained as a p a q (ba)a −p a −q = ba 1−p−q a p−2q = ba 1−n+3q a n ,   b ba 1−n+3q a n b−1 = ba −1+2n−3q a −1+n−3q ,   b2 ba 1−n+3q a n b−2 = ba −n a 1−2n+3q ,

(10.23) (10.24) (10.25)

where we note that these elements conjugate to ba, as well as conjugates of b, do not include ba 2 when N/3 = integer. Therefore, for N/3 integer, the elements ba  a m () are classified into three conjugacy classes CN 2 /3 for  = 0, 1, 2 i.e.,



N −3 () −n−3m n CN = ba a m = 0, 1, . . . , ; n = 0, . . . , N − 1 . 2 /3

3

(10.26)

In the same way, the elements b2 a  a m are classified into three conjugacy classes () CN 2 /3 for  = 0, 1, 2, i.e., () CN 2 /3



N −3 2 −n−3m n ; n = 0, . . . , N − 1 . = b a a m = 0, 1, . . . , 3

(10.27)

92

10

Δ(3N 2 )

We summarize the conjugacy classes of Δ(3N 2 ) for N/3 integer as follows: C1 :

{e},  k −k , k = N3 , 2N : a a 3 , 



 2N N N 2N (,m) , , , , C3 : a  a m , a −+m a − , a −m a −m , (, m) = 3 3 3 3



N −3 (1,p) , n = 0, 1, . . . , N − 1 , ba p−n−3m a n

m = 0, 1, . . . , CN 2 /3 : 3 p = 0, 1, 2,



N −3 (2,p) 2 p−n−3m n

b a a m = 0, 1, . . . , CN 2 /3 : , n = 0, 1, . . . , N − 1 , 3 p = 0, 1, 2. (10.28) C1(k)

The order h of each element in the given conjugacy classes, i.e., such that g h = e, are given as follows: C1 :

h = 1,

(k) C1 : C3(,m) : (1,p) CN 2 /3 : (2,p) CN 2 /3 :

h = 3, h = N/gcd(N, , m),

(10.29)

h = 3, h = 3. (1,p)

(2,p)

(k) (,m) , CN 2 /3 , and CN 2 /3 are 2, (N 2 −3)/3, The numbers of conjugacy classes C1 , C3 3, and 3, respectively. The total number of conjugacy classes is therefore equal to 9 + (N 2 − 3)/3. The relations (2.18) and (2.19) for Δ(3N 2 ) with N/3 integer imply

m1 + 22 m2 + 32 m3 + · · · = 3N 2 ,   m1 + m2 + m3 + · · · = 9 + N 2 − 3 /3,

(10.30) (10.31)

which have the solution (m1 , m3 ) = (9, (N 2 − 3)/3). That is, there are nine singlets and (N 2 − 3)/3 triplets.

10.2.2 Characters and Representations There are nine singlet representations of the group Δ(3N 2 ) with N/3 integer. Their characters must satisfy χα (b) = ωk (k = 0, 1, 2) similarly to the case N/3 = integer. In addition, it is found that χα (a) = χα (a  ) = ω ( = 0, 1, 2). Thus, the nine singlets can be specified by combinations of χα (b) and χα (a), i.e., 1k, (k,  = 0, 1, 2) with χα (b) = ωk and χα (a) = χα (a  ) = ω . These characters are shown in Table 10.2.

10.2

Δ(3N 2 ) with N/3 Integer

93

Table 10.2 Characters of Δ(3N 2 ) for N/3 integer h

χ1r,s

C1

1

1

3

C1(k)

3

1

ρ nk+2mk + ρ nk+mk + ρ −2nk−mk

C3(k,) (1,p)

CN 2 /3

(2,p) CN 2 /3

χ3[n][m]

N gcd(N, k, ) 3

ωs(+m)

ρ mk−n−m + ρ nk−m + ρ −nk−mk+n

ωr+sp

0

3

ω2r+sp

0

The triplet representations are also given similarly to the case with N/3 = integer. That is, the generators of Δ(3N 2 ) with N/3 integer are represented by ⎛ ⎞ ⎛  ⎛ −k− ⎞ ⎞ ρ ρ 0 0 0 0 0 1 0 b = ⎝0 0 1⎠, a = ⎝ 0 ρk a = ⎝ 0 0 ⎠, ρ 0 ⎠ , −k− 1 0 0 0 0 ρ 0 0 ρk (10.32) on the triplet 3[k][] . It should be noted that the matrices a and a  are trivial for the case (k, ) = (0, 0), (N/3, N/3), (2N/3, 2N/3). Thus, we exclude such values of (k, ). These characters are shown in Table 10.2.

10.2.3 Tensor Products For N/3 integer, we present the tensor products of two triplets: ⎛ ⎞ ⎞ ⎛ y ,−k  − x,−k− ⎝ xk, ⎠ and ⎝ yk , ⎠ . x−k−,k 3 y−k  − ,k  3   [k][]

(10.33)

[k ][ ]

(k  ,  ) = [k

+ mN/3][ + mN/3] for m = 0, 1, 2, their tensor products are Unless the same as (10.18). For (k  ,  ) = [−k + mN/3][− + mN/3] (m = 0, 1 or 2), the tensor products of the above triplets are: ⎛ ⎛ ⎞ ⎞ x,−k− y−+mN/3,k+−2mN/3 ⎝ xk, ⎠ ⊗ ⎝ y−k+mN/3,−+mN/3 ⎠ x−k−,k 3 yk+−2mN/3,−k+mN/3 3 [k][]

[−k+mN/3][−+mN/3]

= (x,−k− y−+mN/3,k+−2mN/3 + xk, y−k+mN/3,−+mN/3 + x−k−,k yk+−2mN/3,−k+mN/3 )10,m  ⊕ x,−k− y−+mN/3,k+−2mN/3 + ω2 xk, y−k+mN/3,−+mN/3  + ωx−k−,k yk+−2mN/3,−k+mN/3 1 1,m  ⊕ x,−k− y−+mN/3,k+−2mN/3 + ωxk, y−k+mN/3,−+mN/3

94

10

 + ω2 x−k−,k yk+−2mN/3,−k+mN/3 1 2,m ⎞ ⎛ xk, yk+−2mN/3,−k+mN/3 ⊕ ⎝ x−k−,k y−+mN/3,k+−2mN/3 ⎠ x,−k− y−k+mN/3,−+mN/3 3 [−k−2+mN/3][2k+−2mN/3] ⎛ ⎞ x−k−,k y−k+mN/3,−+mN/3 ⊕ ⎝ x,−k− yk+−2mN/3,−k+mN/3 ⎠ . xk, y−+mN/3,k+−2mN/3 3

Δ(3N 2 )

(10.34)

[k+2−2mN/3][−2k−+mN/3]

The product of 3[k][] and 1r,s is ⎞ ⎛ ⎞ ⎛ x,−k− z x,−k− ⎝ xk, ⎠ ⊗ (z)1r,s = ⎝ ωr xk, z ⎠ x−k−,k 3 ω2r x−k−,k z 3 [k][]

.

(10.35)

[k+sN/3][+sN/3]

The tensor products of the singlets 1k, and 1k  , are 1k, ⊗ 1k , = 1k+k mod 3,+

(10.36)

mod 3 .

10.3 Δ(27) We discuss here a simple example of the group Δ(3N 2 ). The simplest such group, viz., Δ(3), is nothing but Z3 . The next, viz., Δ(12), is isomorphic to A4 . Thus, the simplest non-trivial example is Δ(27). The conjugacy classes of Δ(27) are: C1 : (1) C1 : (2) C1 : C3(0,1) : (0,2) C3 : (1,p) : C3 (2,p) C3 :

{e},  2 aa ,  2  a a ,   a , a, a 2 a 2 ,  2 2 a , a , aa  ,  p ba , ba p−1 a  , ba p−2 a 2 ,  2 p 2 p−1  2 p−2 2 b a ,b a a ,b a a ,

h = 1, h = 3, h = 3, h = 3,

(10.37)

h = 3, h = 3, h = 3.

Δ(27) has nine singlets 1k, (k,  = 0, 1, 2) and two triplets 3[0][1] and 3[0][2] . The characters are shown in Table 10.3. The tensor products between triplets are: ⎛ ⎛ ⎞ ⎞ x1,−1 y1,−1 ⎝ x0,1 ⎠ ⊗ ⎝ y0,1 ⎠ x−1,0 3 y−1,0 3 [0][1] [0][1] ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ x1,−1 y1,−1 x0,1 y−1,0 x−1,0 y0,1 = ⎝ x0,1 y0,1 ⎠ ⊕ ⎝ x−1,0 y1,−1 ⎠ ⊕ ⎝ x1,−1 y−1,0 ⎠ , x−1,0 y−1,0 3 x1,−1 y0,1 3 x0,1 y1,−1 3 [0][2] [0][2] [0][2] (10.38)

References

95

Table 10.3 Characters of Δ(27)

h

χ1k,

χ3[0,1]

C1

1

1

3

3

C1(1)

1

1

3ω2

3ω

C1(2)

1

1

3ω

3ω2

3

ω

0

0

3

ω2

0

0

3

ωk+p

0

0

3

ω2k+p

0

0

C1(0,1) C1(0,2) (1,p) C3 (2,p) C3

⎞ ⎞ ⎛ x2,−2 y2,−2 ⎝ x0,2 ⎠ ⊗ ⎝ y0,2 ⎠ x−2,0 3 y−2,0 3 [0][2] [0][2] ⎞ ⎞ ⎛ ⎛ x2,−2 y2,−2 x0,2 y−2,0 = ⎝ x0,2 y0,2 ⎠ ⊕ ⎝ x−2,0 y2,−2 ⎠ x−2,0 y−2,0 3 x2,−2 y0,2 3

χ3[0,2]

⎛

[0][1]

[0][1]

⎞ x−2,0 y0,2 ⊕ ⎝ x2,−2 y−2,0 ⎠ x0,2 y2,−2 3 ⎛

,

[0][1]

(10.39)

⎞ ⎞ ⎛ x1,−1 y−1,1 ⎝ x0,1 ⎠ ⊗ ⎝ y0,−1 ⎠ x−1,0 3 y1,0 3 [0][1] [0][2]   x1,−1 y−1,1 + ω2r x0,1 y0,−1 + ωr x−1,0 y1,0 1 = ⎛

(r,0)

r

  x1,−1 y0,−1 + ω2r x0,1 y1,0 + ωr x−1,0 y−1,1 1 ⊕

(r,1)

r

⊕

  x1,−1 y1,0 + ω2r x0,1 y−1,1 + ωr x−1,0 y0,−1 1

(r,2)

r

.

The tensor products between singlets and triplets are: ⎞ ⎛ ⎞ ⎛ x(1,−1) z x(1,−1) ⎝ x(0,1) ⎠ ⊗ (z)1k, = ⎝ ωr x(0,1) z ⎠ , x(−1,0) 3 ω2r x(−1,0) z 3 ⎞ [0][1] ⎛ ⎛ ⎞ [][1+] x(2,−2) z x(2,−2) ⎝ x(0,2) ⎠ ⊗ (z)1k, = ⎝ ωr x(0,2) z ⎠ . x(−2,0) 3 ω2r x(−2,0) z 3 [0][2]

(10.40)

(10.41)

[][2+]

The tensor products of singlets are easily obtained from (10.36).

References 1. Luhn, C., Nasri, S., Ramond, P.: J. Math. Phys. 48, 073501 (2007). arXiv:hep-th/0701188

Chapter 11

TN

11.1 Generic Aspects We now study the group TN , which is isomorphic to ZN  Z3 (see, e.g., [1–3]). Here we focus on the case where N is any prime number except 3 or any power of such a prime number, i.e., N = p q with p = 3 a prime number and q a positive number. We denote the generators of ZN and Z3 by a and b, respectively. These satisfy a N = e,

b3 = e.

(11.1)

Because of the semi-direct product structure, we impose ba = a m b,

(11.2)

with m = 0. When m = 1 mod N , a and b commute and the group is just the direct product ZN × Z3 . Thus, we impose m = 1 mod N . For example, the case with N = 2 is excluded, because we have only m = 1 mod N for N = 2, except if m = 0. We find     2 (11.3) a = b2 a m b, a m = b2 a m b · · · b2 a m b = b2 a m b = bab2 ,     2 2 2 3 a = ba m b2 , a m = ba m b2 · · · ba m b2 = ba m b2 = bab2 . (11.4) The consistency of these equations implies that 3

and hence,

a m = a,

(11.5)

  m3 − 1 = (m − 1) m2 + m + 1 = 0 mod N.

(11.6)

Here, we focus on the case where the condition m2 + m + 1 = 0 mod N

(11.7)

is satisfied with 1 < m < N . Suppose now that m = 3 with integer . Thus, it follows that m2 + m + 1 = 3(3 + 1) + 1, H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_11, © Springer-Verlag Berlin Heidelberg 2012

(11.8) 97

98 Table 11.1 m for N ≤ 50

11

TN

N

7

13

19

31

43

49 (= 7 × 7)

m

2

3

7

5

6

18

and 3(3 + 1) is always a multiple of 6, i.e., 3(3 + 1) = 6k and m2 + m + 1 = 6k + 1. Similarly, when m = 3 + 2, we obtain m2 + m + 1 = 3(3 + 5) + 7.

(11.9)

Then, it is found that m2 + m + 1 = 6k + 1. These results show that the possible values of N have the form N = 6k + 1. On the other hand, when m = 3 + 1, we find   m2 + m + 1 = 3 3( + 1) + 1 . (11.10) Here, ( + 1) is always even, i.e., ( + 1) = 2 , and we can write   m2 + m + 1 = 3 6 + 1 .

(11.11)

This implies that, for N = pq (p = 3), the possible values of N would be N = 6 + 1. If we took N = 3, we could not find non-trivial m, because m3 − 1 = 7 for m = 2. We thus find that the possible values of N are also N = 6k + 1 for m = 3 + 1. Indeed, explicit computation with (11.7) leads to the following possible values of N :   N = 7, 13, 19, 31, 43, 49(= 7 × 7) for N ≤ 50. (11.12) These values and the corresponding values of m are shown in Table 11.1. All elements of TN can be expressed in terms of the two generators a and b as follows: g = bk a  ,

(11.13)

for k = 0, 1, 2, and  = 0, . . . , N − 1. The generators b and a are represented, e.g., by ⎞ ⎛ ⎞ ⎛ ρ 0 0 0 1 0 0 ⎠, (11.14) b = ⎝0 0 1⎠, a = ⎝ 0 ρm 2 1 0 0 0 0 ρm where we define ρ = e2πi/N .

11.1.1 Conjugacy Classes All elements bk a  of TN are classified into 3 + (N − 1)/3 conjugacy classes:

11.1

Generic Aspects

C1 : (1) CN (2) CN

: :

C3[k] :

99

{e}, h = 1, 

N−2 N−1 , h = 3, b, ba, . . . , ba , ba 

2 2 b , b a, . . . , b2 a N−2 , b2 a N−1 , h = 3,

k km km2  , h = N/gcd(N, k), a ,a ,a

(11.15)

where gcd(N, k) = N when N is a prime number. The relations (2.18) and (2.19) lead to m1 + m2 + m3 = 3 + (N − 1)/3, m1 + 4m2 + 9m3 = 3N.

(11.16) (11.17)

For specific values of N in (11.12), their solutions are found to be m1 = 3, m2 = 0, and m3 = (N − 1)/3.

11.1.2 Characters and Representations The group TN has 3 singlets. Since b3 = e, the characters of the three singlets have three possible values χ1k (b) = ωk with k = 0, 1, 2, and they correspond to three singlets 1k . Note that χ1k (a) = 1, because χ1k (b) = χ1k (ba). These characters are shown in Table 11.2. Now consider the triplets. We use the notation [k] defined by [k] = k, km, or km2 (mod N ).

(11.18)

We also define ξ[k] = ρ k + ρ km + ρ . The notation ξ¯[k] is defined as the complex conjugate of ξ[k] , which is ξ[N−k] . The two generators b and a are represented, e.g., by ⎛ ⎞ ⎞ ⎛ k 0 0 ρ 0 1 0 0 ⎠, b = ⎝0 0 1⎠, (11.19) a = ⎝ 0 ρ km 2 km 1 0 0 0 0 ρ km2

on the triplets 3[k] . We also define the vector 3[k] by ⎛ ⎞ xk 3[k] ≡ ⎝ xkm ⎠ , xkm2

(11.20)

for k ∈ N − 1, where k corresponds to the ZN charge. Thus we have (N − 1)/3 different triplets. The characters are shown in Table 11.2. Note also that we use the notation 3¯ [k] = 3[N−k] .

11.1.3 Tensor Products The tensor products between triplets are obtained in general as follows:

100

11

Table 11.2 Characters of TN

χ 11

χ 12

TN

h

χ1 0

χ3[]

C1

1

1

1

1

3

(1) CN

3

1

ω

ω2

0

(2) CN

3

1

ω2

ω

0

C3[k]

N/gcd(N, k)

1

1

1

ξ[k]

⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ y xk y xk ym xk ⎝ xkm ⎠ ⊗ ⎝ ym ⎠ = ⎝ xkm ym ⎠ ⊕ ⎝ xkm ym2 ⎠ xkm2 3 ym2 3 xkm2 ym2 3 xkm2 y 3 [k] [] [k+] [k+m] ⎛ ⎞ xk ym2 ⊕ ⎝ xkm y ⎠ , (11.21) xkm2 ym 3 ⎛

[k+m2 ]

for [k] = [N − ]. When [k] = [N − ], one of 3[k+] , 3[k+m] , or 3[k+m2 ] can be reduced to three singlets. For example, if k +  = 0 mod N , the triplet 3[k+] can be reduced as follows: ⎞ ⎛ x k y     ⎝ xkm ym ⎠ xk y + ω2k xkm ym + ωk xkm2 ym2 1  . (11.22) =⊕ k xkm2 ym2 3 k  =0,1,2 [k+]

Similarly, when k + mn = 0 mod N for n = 1, 2, the triplet 3[k+mn ] can be reduced to three singlets. In addition, the tensor products including singlets are simply 1k ⊗ 3[] = 3[] ,

1k ⊗ 1k  = 1k+k  .

(11.23)

11.2 T7 Here we study the smallest group TN , that is, T7 . We denote the generator of Z7 by a and that of Z3 by b. They satisfy a 7 = 1,

ba = a 2 b.

(11.24)

Using these, all elements of T7 can be written in the form g = bk a  , with k = 0, 1, 2, and  = 0, . . . , 6. The generators a and b are represented, e.g., by ⎛ ⎞ ⎛ ρ 0 1 0 b = ⎝0 0 1⎠, a =⎝0 1 0 0 0

(11.25)

0 ρ2 0

⎞ 0 0 ⎠, ρ4

where ρ = e2iπ/7 . These elements are classified into five conjugacy classes:

(11.26)

11.2

101

T7

Table 11.3 Characters of T7

C1 : (1)

C7 : (2) C7

C3 : C3¯ :

:

h

χ1 0

χ1 1

χ 12

C1

1

1

1

1

3

3

C7(1)

3

1

ω

ω2

0

0

C7(2)

3

1

ω2

ω

0

C3

7

1

1

1

0 ξ¯

C3¯

7

1

1

1

ξ ξ¯

{e}, 

b, ba, ba 2 , ba 3 , ba 4 , ba 5 , ba 6 , 

2 2 b , b a, b2 a 2 , b2 a 3 , b2 a 4 , b2 a 5 , b2 a 6 , 

a, a 2 , a 4 ,

3 5 6 a ,a ,a ,

χ3

χ3¯

ξ

h = 1, h = 3, h = 3,

(11.27)

h = 7, h = 7.

¯ The and two triplets 3 and 3. T7 has three singlet representations 1k with k = 0, 1, 2, √ characters are shown in Table 11.3, where ξ = (−1 + i 7)/2. Using the order of ρ in a, we define the triplets 3 and 3¯ by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x−1 x6 x1 3¯ ≡ ⎝ x−2 ⎠ = ⎝ x5 ⎠ . (11.28) 3 ≡ ⎝ x2 ⎠ , x4 x−4 x3 The tensor products between triplets are ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ x1 y1 x2 y4 x4 y2 x4 y4 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x4 y1 ⎠ ⊕ ⎝ x1 y4 ⎠ ⊕ ⎝ x1 y1 ⎠ , x4 y4 x1 y2 3¯ x2 y1 3¯ x2 y2 3 ⎛ ⎞3 ⎛ ⎞3 ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ x6 y6 x5 y3 x3 y5 x3 y3 ⎝ x5 ⎠ ⊗ ⎝ y5 ⎠ = ⎝ x3 y6 ⎠ ⊕ ⎝ x6 y3 ⎠ ⊕ ⎝ x6 y6 ⎠ , x3 ¯ y3 ¯ x6 y5 3 x5 y6 3 x5 y5 3¯ ⎛ ⎞3 ⎛ ⎞3 ⎛ ⎛ ⎞ ⎞ x1 y6 x2 y6 x1 y5 ⎝ x2 ⎠ ⊗ ⎝ y5 ⎠ = ⎝ x4 y5 ⎠ ⊕ ⎝ x2 y3 ⎠ x4 3 y3 3¯ x1 y3 3 x4 y6 3¯   x1 y6 + ω2k x2 y5 + ωk x4 y3 1 . ⊕ k

(11.29)

(11.30)

(11.31)

k=0,1,2

The tensor products between singlets are (x)10 (y)10 = (x)11 (y)12 = (x)12 (y)11 = (xy)10 , (x)11 (y)11 = (xy)12 , (x)12 (y)12 = (xy)11 .

(11.32)

The tensor products between triplets and singlets are 1k ⊗ 3 = 3,

¯ 1k ⊗ 3¯ = 3,

where k = 0, 1, 2. Notice that singlets have no affect on the form of triplets.

(11.33)

102

11

Table 11.4 Characters of T13 . ξ¯i is defined as the complex conjugate of ξi

TN

h

χ 10

χ1 1

χ1 2

χ3 1

χ3¯ 1

χ3 2

χ3¯ 2

C1

1

1

1

1

3

3

3

3

(1) C13

3

1

ω

ω2

0

0

0

0

0 ξ¯1

0

0 ξ¯2

(2) C13

3

1

ω2

ω

0

C 31

13

1

1

1

C3¯ 1

13

1

1

1

ξ1 ξ¯1

C 32

13

1

1

1

C3¯ 2

13

1

1

1

ξ1 ξ¯2

ξ2 ξ¯2

ξ2

ξ2 ξ¯2

ξ2 ξ¯1

ξ1 ξ¯1

ξ1

11.3 T13 The non-Abelian discrete group T13 is isomorphic to Z13  Z3 [3, 4]. This group is a subgroup of SU(3), and known to be the minimal non-Abelian discrete group with two complex triplets as irreducible representations. We denote the generators of Z13 and Z3 by a and b, respectively. They satisfy a 13 = 1,

ba = a 3 b.

(11.34)

Using these, all elements of T13 can be expressed in the form g = bk a  , with k = 0, 1, 2, and  = 0, . . . , 12. The generators a and b are represented, e.g., by ⎛ ⎞ ⎛ ρ 0 1 0 b = ⎝0 0 1⎠, a =⎝0 1 0 0 0

(11.35)

0 ρ3 0

⎞ 0 0 ⎠, ρ9

(11.36)

where ρ = e2iπ/13 . These elements are classified into seven conjugacy classes: C1 : (1)

C13 : (2) C13

C31 : C3¯ 1 : C32 : C3¯ 2 :

:

{e}, 

b, ba, ba 2 , . . . , ba 10 , ba 11 , ba 12 , 

2 2 b , b a, b2 a 2 , . . . , b2 a 10 , b2 a 11 , b2 a 12 , 

a, a 3 , a 9 ,

4 10 12  a ,a ,a ,

2 5 6 a ,a ,a ,

7 8 11  a ,a ,a ,

h = 1, h = 3, h = 3, h = 13,

(11.37)

h = 13, h = 13, h = 13.

T13 has three singlet representations 1k with k = 0, 1, 2, together with two complex triplets 31 and 32 and their conjugates as irreducible representations. The characters are shown in Table 11.4, where ξ1 ≡ ρ + ρ 3 + ρ 9 , ξ2 ≡ ρ 2 + ρ 5 + ρ 6 , and ω ≡ e2iπ/3 .

11.3

103

T13

Next we consider the multiplication rules of the group T13 . We define the triplets by

⎞ x1 31 ≡ ⎝ x3 ⎠ , x9

⎞ x¯12 3¯ 1 ≡ ⎝ x¯10 ⎠ , x¯4

⎛

⎛

⎞ y2 32 = ⎝ y6 ⎠ , y5

⎞ y¯11 3¯ 2 ≡ ⎝ y¯7 ⎠ , (11.38) y¯8

⎛

⎛

where the subscripts denote the Z13 charge of each element. The tensor products between triplets are: ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ x1 y1 x3 y9 x9 y3 x1 y1 ⎝ x3 ⎠ ⊗ ⎝ y3 ⎠ = ⎝ x9 y1 ⎠ ⊕ ⎝ x1 y9 ⎠ ⊕ ⎝ x3 y3 ⎠ , (11.39) x9 3 y9 3 x1 y3 3¯ x3 y1 3¯ x9 y9 3 1 1 1 1 2 ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ ⎛ x¯12 y¯12 x¯10 y¯4 x¯4 y¯10 x¯12 y¯12 ⎝ x¯10 ⎠ ⊗ ⎝ y¯10 ⎠ = ⎝ x¯4 y¯12 ⎠ ⊕ ⎝ x¯12 y¯4 ⎠ ⊕ ⎝ x¯10 y¯10 ⎠ , x¯4 3¯ y¯4 3¯ x¯12 y¯10 3 x¯10 y¯12 3 x¯4 y¯4 3¯ 1

⎛

1

⎛

⎞

1

2

1

(11.40)

⎞

x1 y¯12   ⎝ x3 ⎠ ⊗ ⎝ y¯10 ⎠ = x1 y¯12 + ω2k x3 y¯10 + ωk x9 y¯4 1 k x9 3 y¯4 3¯ k=0,1,2 1 1 ⎛ ⎛ ⎞ ⎞ x3 y¯12 x1 y¯10 ⊕ ⎝ x9 y¯10 ⎠ ⊕ ⎝ x3 y¯4 ⎠ , (11.41) x1 y¯4 3 x9 y¯12 3¯ 2 2 ⎛ ⎞ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ y2 x5 y6 x6 y5 x6 y6 x2 ⎝ x6 ⎠ ⊗ ⎝ y6 ⎠ = ⎝ x2 y5 ⎠ ⊕ ⎝ x5 y2 ⎠ ⊕ ⎝ x5 y5 ⎠ , (11.42) x5 3 y5 3 x6 y2 3¯ x2 y6 3¯ x2 y2 3¯ 2 2 2 2 1 ⎛ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ y¯11 x¯8 y¯7 x¯7 y¯8 x¯7 y¯7 x¯11 ⎝ x¯7 ⎠ ⊗ ⎝ y¯7 ⎠ = ⎝ x¯11 y¯8 ⎠ ⊕ ⎝ x¯8 y¯11 ⎠ ⊕ ⎝ x¯8 y¯8 ⎠ , (11.43) x¯8 3¯ y¯8 3¯ x¯7 y¯11 3 x¯11 y¯7 3 x¯11 y¯11 3 2 2 2 2 1 ⎛ ⎞ ⎞ ⎛ x2 y¯11   ⎝ x6 ⎠ ⊗ ⎝ y¯7 ⎠ = x2 y¯11 + ω2k x6 y¯7 + ωk x5 y¯8 1 k x5 3 y¯8 3¯ k=0,1,2 2 2 ⎞ ⎞ ⎛ ⎛ x6 y¯8 x5 y¯7 ⊕ ⎝ x5 y¯11 ⎠ ⊕ ⎝ x2 y¯8 ⎠ , (11.44) x2 y¯7 3 x6 y¯11 3¯ 1 1 ⎛ ⎞ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ y2 x9 y6 x9 y2 x9 y5 x1 ⎝ x3 ⎠ ⊗ ⎝ y6 ⎠ = ⎝ x1 y5 ⎠ ⊕ ⎝ x1 y6 ⎠ ⊕ ⎝ x1 y2 ⎠ , (11.45) x9 3 y5 3 x3 y2 3 x3 y5 3¯ x3 y6 3 2 1 2 2 1 ⎛ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ y¯11 x1 y¯11 x3 y¯8 x3 y¯11 x1 ⎝ x3 ⎠ ⊗ ⎝ y¯7 ⎠ = ⎝ x3 y¯7 ⎠ ⊕ ⎝ x9 y¯11 ⎠ ⊕ ⎝ x9 y¯7 ⎠ , (11.46) x9 3 y¯8 3¯ x9 y¯8 3¯ x1 y¯7 3¯ x1 y¯8 3 1

2

1

2

1

104

11

TN

⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ y¯12 x2 y¯12 x2 y¯10 x5 y¯10 x2 ⎝ x6 ⎠ ⊗ ⎝ y¯10 ⎠ = ⎝ x6 y¯10 ⎠ ⊕ ⎝ x6 y¯4 ⎠ ⊕ ⎝ x2 y¯4 ⎠ , (11.47) x5 3 y¯4 3¯ x5 y¯4 3 x5 y¯12 3¯ x6 y¯12 3 2 1 1 1 2 ⎛ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ y¯11 x¯4 y¯8 x¯4 y¯7 x¯4 y¯11 x¯12 ⎝ x¯10 ⎠ ⊗ ⎝ y¯7 ⎠ = ⎝ x¯12 y¯11 ⎠ ⊕ ⎝ x¯12 y¯8 ⎠ ⊕ ⎝ x¯12 y¯7 ⎠ . x¯4 3¯ y¯8 3¯ x¯10 y¯7 3¯ x¯10 y¯11 3¯ x¯10 y¯8 3 ⎛

1

2

1

2

2

(11.48) The tensor products between singlets are (x)10 (y)10 = (x)11 (y)12 = (x)12 (y)11 = (xy)10 , (x)11 (y)11 = (xy)12 , (x)12 (y)12 = (xy)11 .

(11.49)

The tensor products between triplets and singlets are 1j ⊗ 3k = 3k ,

(11.50)

where j and k run over all singlets and triplets, respectively. Note that singlets do not affect the form of triplets.

11.4 T19 The non-Abelian discrete group T19 is isomorphic to Z19  Z3 . This group is a subgroup of SU(3), and known to be the minimal non-Abelian discrete group with three complex triplets as irreducible representations. We denote the generators of Z19 and Z3 by a and b, respectively. They satisfy a 19 = 1,

ba = a 7 b.

(11.51)

Using these, all elements of T19 can be expressed in the form g = bk a  ,

(11.52)

with k = 0, 1, 2, and  = 0, . . . , 18. The generators a and b are represented, e.g., by ⎛

0 b = ⎝0 1

1 0 0

⎞ 0 1⎠, 0

⎛

ρ a =⎝0 0

0 ρ7 0

⎞ 0 0 ⎠, ρ 11

(11.53)

11.4

105

T19

Table 11.5 Characters of T19 . ξ¯i is defined as the complex conjugate of ξi

h

χ 10

χ 11

χ1 2

χ3 1

χ3¯ 1

χ 32

χ3¯ 2

χ3 3

χ3¯ 3

C1

1

1

1

1

3

3

3

3

3

3

(1) C19

3

1

ω

ω2

0

0

0

0

0

0

0 ξ¯1

0

0 ξ¯2

0

0 ξ¯3

(2) C19

3

1

ω2

ω

0

C 31

19

1

1

1

C3¯ 1

19

1

1

1

ξ1 ξ¯1

C 32

19

1

1

1

C3¯ 2

19

1

1

1

C 33

19

1

1

1

C3¯ 3

19

1

1

1

ξ2 ξ¯2 ξ3 ξ¯3

ξ1 ξ¯2 ξ2 ξ¯3 ξ3

ξ2 ξ¯2 ξ3 ξ¯3 ξ1 ξ¯1

ξ2 ξ¯3 ξ3 ξ¯1 ξ1

ξ3 ξ¯3 ξ1 ξ¯1 ξ2 ξ¯2

ξ3 ξ¯1 ξ1 ξ¯2 ξ2

where ρ = e2iπ/19 . These elements are classified into nine conjugacy classes: C1 : (1) C19 (2) C19

C31 : C3¯ 1 : C32 : C3¯ 2 : C33 : C3¯ 3 :

: :

{e}, 

b, ba, ba 2 , . . . , ba 16 , ba 17 , ba 18 , 

2 2 b , b a, b2 a 2 , . . . , b2 a 16 , b2 a 17 , b2 a 18 , 

a, a 7 , a 11 ,

8 12 18  a ,a ,a ,

2 3 14  a ,a ,a ,

5 16 17  a ,a ,a ,

4 6 9 a ,a ,a ,

10 13 15  a ,a ,a ,

h = 1, h = 3, h = 3, h = 19, h = 19,

(11.54)

h = 19, h = 19, h = 19, h = 19.

T19 has three singlet representations 1k with k = 0, 1, 2, and three complex triplets 31 , 32 , and 33 , together with their conjugates, as irreducible representations. The characters are shown in Table 11.5, where ξ1 ≡ ρ + ρ 7 + ρ 11 , ξ2 ≡ ρ 2 + ρ 3 + ρ 14 , ξ3 ≡ ρ 4 + ρ 6 + ρ 9 , and ω ≡ e2iπ/3 . Next we consider the multiplication rules of the T19 group. We define the triplets by ⎛

⎞ x1 31 ≡ ⎝ x7 ⎠ , x11 ⎛ ⎞ y¯17 3¯ 2 ≡ ⎝ y¯5 ⎠ , y¯16

⎛

⎞ x¯18 3¯ 1 ≡ ⎝ x¯12 ⎠ , x¯8 ⎛ ⎞ z4 33 = ⎝ z9 ⎠ , z6

⎛

⎞ y2 32 = ⎝ y14 ⎠ , y3 ⎛ ⎞ z¯ 15 3¯ 3 ≡ ⎝ z¯ 10 ⎠ , z¯ 13

where the subscripts denote the Z19 charge of each element. The tensor products between triplets are:

(11.55)

106

11

TN

⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ y1 x7 y11 x11 y7 x1 y 1 x1 ⎝ x7 ⎠ ⊗ ⎝ y7 ⎠ = ⎝ x11 y1 ⎠ ⊕ ⎝ x1 y11 ⎠ ⊕ ⎝ x7 y7 ⎠ , (11.56) x11 3 y11 3 x1 y7 3¯ x7 y1 3¯ x11 y11 3 ⎛ ⎛ ⎛ ⎞1 ⎛ ⎞1 ⎛ ⎞1 ⎞1 ⎞2 x¯18 y¯18 x¯8 y¯12 x¯12 y¯8 x¯18 y¯18 ⎝ x¯12 ⎠ ⊗ ⎝ y¯12 ⎠ = ⎝ x¯18 y¯8 ⎠ ⊕ ⎝ x¯8 y¯18 ⎠ ⊕ ⎝ x¯12 y¯12 ⎠ , x¯8 3¯ y¯8 3¯ x¯12 y¯18 3 x¯18 y¯12 3 x¯8 y¯8 3¯ ⎛

1

⎛

1

⎛

⎞

1

2

1

(11.57)

⎞

x1 y¯18   ⎝ x7 ⎠ ⊗ ⎝ y¯12 ⎠ = x1 y¯18 + ω2k x7 y¯12 + ωk x11 y¯8 1 k x11 3 y¯8 3¯ k=0,1,2 1 1 ⎞ ⎞ ⎛ ⎛ x11 y¯12 x7 y¯8 ⊕ ⎝ x1 y¯8 ⎠ ⊕ ⎝ x11 y¯18 ⎠ , (11.58) x7 y¯18 3 x1 y¯12 3¯ 3 3 ⎛ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ x2 y2 x2 y2 x3 y14 x14 y3 ⎝ x14 ⎠ ⊗ ⎝ y14 ⎠ = ⎝ x14 y14 ⎠ ⊕ ⎝ x2 y3 ⎠ ⊕ ⎝ x3 y2 ⎠ , (11.59) x3 3 y3 3 x3 y3 3 x14 y2 3¯ x2 y14 3¯ 2 2 3 2 2 ⎞ ⎞ ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ ⎛ ⎛ x¯17 y¯17 x¯17 y¯17 x¯5 y¯16 x¯16 y¯5 ⎝ x¯5 ⎠ ⊗ ⎝ y¯5 ⎠ = ⎝ x¯5 y¯5 ⎠ ⊕ ⎝ x¯16 y¯17 ⎠ ⊕ ⎝ x¯17 y¯16 ⎠ , x¯16 3¯ y¯16 3¯ x¯16 y¯16 3¯ x¯17 y¯5 3 x¯5 y¯17 3 2

⎛

⎞

2

⎛

3

2

2

(11.60)

⎞

y¯17 x2   ⎝ x14 ⎠ ⊗ ⎝ y¯5 ⎠ = x2 y¯17 + ω2k x14 y¯5 + ωk x3 y¯16 1 k x3 3 y¯16 3¯ k=0,1,2 2 2 ⎛ ⎛ ⎞ ⎞ x3 y¯17 x2 y¯16 ⊕ ⎝ x2 y¯5 ⎠ ⊕ ⎝ x14 y¯17 ⎠ , (11.61) x14 y¯16 3 x3 y¯5 3¯ ⎞ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎛1 ⎛1 ⎞ x4 y4 x9 y9 x6 y9 x9 y6 ⎝ x9 ⎠ ⊗ ⎝ y9 ⎠ = ⎝ x6 y6 ⎠ ⊕ ⎝ x4 y6 ⎠ ⊕ ⎝ x6 y4 ⎠ , (11.62) x6 3 y6 3 x4 y4 3¯ x9 y4 3¯ x4 y9 3¯ 1 3 3 3 3 ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ ⎛ y¯15 x¯10 y¯10 x¯10 y¯13 x¯13 y¯10 x¯15 ⎝ x¯10 ⎠ ⊗ ⎝ y¯10 ⎠ = ⎝ x¯13 y¯13 ⎠ ⊕ ⎝ x¯13 y¯15 ⎠ ⊕ ⎝ x¯15 y¯13 ⎠ , x¯13 3¯ y¯13 3¯ x¯15 y¯15 3 x¯15 y¯10 3 x¯10 y¯15 3 3

3

1

3

3

⎛

⎛ ⎞ ⎞ x4 y¯15   ⎝ x9 ⎠ ⊗ ⎝ y¯10 ⎠ = x4 y¯15 + ω2k x9 y¯10 + ωk x6 y¯13 1 k x6 3 y¯13 3¯ k=0,1,2 3 3 ⎛ ⎛ ⎞ ⎞ x6 y¯15 x4 y¯13 ⊕ ⎝ x4 y¯10 ⎠ ⊕ ⎝ x9 y¯15 ⎠ , x9 y¯13 3 x6 y¯10 3¯ 2

2

(11.63)

(11.64)

11.4

107

T19

⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ y2 x7 y14 x1 y3 x1 y14 x1 ⎝ x7 ⎠ ⊗ ⎝ y14 ⎠ = ⎝ x11 y3 ⎠ ⊕ ⎝ x7 y2 ⎠ ⊕ ⎝ x7 y3 ⎠ , x11 3 y3 3 x1 y2 3 x11 y14 3 x11 y2 3¯ ⎛

1

2

2

3

3

(11.65) ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ x1 y¯17 x1 y¯17 x7 y¯16 x1 y¯16 ⎝ x7 ⎠ ⊗ ⎝ y¯5 ⎠ = ⎝ x7 y¯5 ⎠ ⊕ ⎝ x11 y¯17 ⎠ ⊕ ⎝ x7 y¯17 ⎠ , x11 3 y¯16 3¯ x11 y¯16 3¯ x1 y¯5 3 x11 y¯5 3¯ ⎛

1

⎛

2

⎛

⎞

1

⎛

⎞

3

⎛

⎞

2

⎛

⎞

(11.66)

⎞

x1 y4 x11 y6 x11 y4 x11 y9 ⎝ x7 ⎠ ⊗ ⎝ y9 ⎠ = ⎝ x1 y4 ⎠ ⊕ ⎝ x1 y9 ⎠ ⊕ ⎝ x1 y6 ⎠ , x11 3 y6 3 x7 y9 3¯ x7 y6 3¯ x7 y4 3 1

3

2

3

1

(11.67) ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ x1 y¯15 x7 y¯10 x7 y¯13 x11 y¯10 ⎝ x7 ⎠ ⊗ ⎝ y¯10 ⎠ = ⎝ x11 y¯13 ⎠ ⊕ ⎝ x11 y¯15 ⎠ ⊕ ⎝ x1 y¯13 ⎠ , x11 3 y¯13 3¯ x1 y¯15 3¯ x1 y¯10 3 x7 y¯15 3 ⎛

1

⎛

3

⎛

⎞

2

⎛

⎞

1

⎛

⎞

2

⎛

⎞

(11.68)

⎞

x2 y¯18 x2 y¯18 x3 y¯18 x3 y¯12 ⎝ x14 ⎠ ⊗ ⎝ y¯12 ⎠ = ⎝ x14 y¯12 ⎠ ⊕ ⎝ x2 y¯12 ⎠ ⊕ ⎝ x2 y¯8 ⎠ , x3 3 y¯8 3¯ x3 y¯8 3 x14 y¯8 3 x14 y¯18 3¯ 2

1

1

2

3

(11.69) ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ x¯18 y¯17 x¯12 y¯5 x¯18 y¯5 x¯18 y¯16 ⎝ x¯12 ⎠ ⊗ ⎝ y¯5 ⎠ = ⎝ x¯8 y¯16 ⎠ ⊕ ⎝ x¯12 y¯16 ⎠ ⊕ ⎝ x¯12 y¯17 ⎠ , x¯8 3¯ y¯16 3¯ x¯18 y¯17 3¯ x¯8 y¯17 3 x¯8 y¯5 3¯ ⎛

1

⎛

2

⎛

⎞

2

⎛

⎞

3

⎛

⎞

3

⎛

⎞

(11.70)

⎞

y¯18 x9 y¯12 x9 y¯8 x6 y¯12 x4 ⎝ x9 ⎠ ⊗ ⎝ y¯12 ⎠ = ⎝ x6 y¯8 ⎠ ⊕ ⎝ x6 y¯18 ⎠ ⊕ ⎝ x4 y¯8 ⎠ , x6 3 y¯8 3¯ x4 y¯18 3 x4 y¯12 3¯ x9 y¯18 3¯ 3

⎛

1

⎛

⎞

2

⎛

⎞

2

⎛

⎞

1

⎛

⎞

⎞

(11.71)

y¯15 x¯8 y¯13 x¯8 y¯15 x¯8 y¯10 x¯18 ⎝ x¯12 ⎠ ⊗ ⎝ y¯10 ⎠ = ⎝ x¯18 y¯15 ⎠ ⊕ ⎝ x¯18 y¯10 ⎠ ⊕ ⎝ x¯18 y¯13 ⎠ , x¯8 3¯ y¯13 3¯ x¯12 y¯10 3 x¯12 y¯13 3 x¯12 y¯15 3¯ 1

3

2

3

1

⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ y4 x14 y9 x14 y6 x14 y4 x2 ⎝ x14 ⎠ ⊗ ⎝ y9 ⎠ = ⎝ x3 y6 ⎠ ⊕ ⎝ x3 y4 ⎠ ⊕ ⎝ x3 y9 ⎠ , x3 3 y6 3 x2 y4 3 x2 y9 3 x2 y6 3¯ ⎛

2

⎛

3

⎛

⎞

3

⎛

⎞

1

⎛

⎞

(11.72)

1

⎛

⎞

⎞

(11.73)

y¯15 x2 y¯15 x3 y¯15 x2 y¯13 x2 ⎝ x14 ⎠ ⊗ ⎝ y¯10 ⎠ = ⎝ x14 y¯10 ⎠ ⊕ ⎝ x2 y¯10 ⎠ ⊕ ⎝ x14 y¯15 ⎠ , x3 3 y¯13 3¯ x3 y¯13 3¯ x14 y¯13 3¯ x3 y¯10 3¯ 2

3

2

1

3

(11.74)

108

11

TN

⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ ⎞ y¯17 x4 y¯17 x6 y¯17 x4 y¯16 x4 ⎝ x9 ⎠ ⊗ ⎝ y¯5 ⎠ = ⎝ x9 y¯5 ⎠ ⊕ ⎝ x4 y¯5 ⎠ ⊕ ⎝ x9 y¯17 ⎠ , (11.75) x6 3 y¯16 3¯ x6 y¯16 3 x9 y¯16 3 x6 y¯5 3 1 ⎛ ⎛ ⎛ ⎞3 ⎛ ⎞2 ⎛ ⎞2 ⎞3 ⎞ x¯17 y¯15 x¯5 y¯10 x¯5 y¯13 x¯5 y¯15 ⎝ x¯5 ⎠ ⊗ ⎝ y¯10 ⎠ = ⎝ x¯16 y¯13 ⎠ ⊕ ⎝ x¯16 y¯15 ⎠ ⊕ ⎝ x¯16 y¯10 ⎠ . x¯16 3¯ y¯13 3¯ x¯17 y¯15 3¯ x¯17 y¯10 3¯ x¯17 y¯13 3 ⎛

2

3

3

1

1

(11.76) The tensor products between singlets are (x)10 (y)10 = (x)11 (y)12 = (x)12 (y)11 = (xy)10 , (x)11 (y)11 = (xy)12 , (x)12 (y)12 = (xy)11 .

(11.77)

The tensor products between triplets and singlets are 1j ⊗ 3k = 3k ,

(11.78)

where j and k run over all the singlets and triplets, respectively. Notice that singlets do not affect the form of triplets.

References 1. 2. 3. 4.

Bovier, A., Luling, M., Wyler, D.: J. Math. Phys. 22, 1536 (1981) Bovier, A., Luling, M., Wyler, D.: J. Math. Phys. 22, 1543 (1981) Fairbairn, W.M., Fulton, T.: J. Math. Phys. 23, 1747 (1982) King, S.F., Luhn, C.: J. High Energy Phys. 0910, 093 (2009). arXiv:0908.1897 [hep-ph]

Chapter 12

Σ(3N 3 )

12.1 Generic Aspects In this chapter, we study Σ(3N 3 ) [1]. This discrete group is defined as a closed  , and Z  , which commute algebra of three Abelian symmetries, namely, ZN , ZN N  , with each other, and their Z3 permutations. Let us denote the generators of ZN , ZN  by a, a  , and a  , respectively, and the Z generator by b. All elements of and ZN 3 Σ(3N 3 ) can be expressed in the form g = bk a m a n a  ,

(12.1)

with k = 0, 1, 2, and m, n,  = 0, . . . , N − 1, where a, a  , a  , and b satisfy a N = a N = a N = 1, b3 = 1,

b2 ab = a  ,

aa  = a  a, b2 a  b = a,

aa  = a  a,

a  a  = a  a  ,

b2 a  b = a  .

These generators a, a  , a  , and b are represented, e.g., by ⎛ ⎞ ⎛ ⎞ 0 1 0 1 0 0 b = ⎝0 0 1⎠, a = ⎝0 1 0 ⎠, 1 0 0 0 0 ρ ⎛ ⎞ ⎛ ⎞ 1 0 0 ρ 0 0 a = ⎝ 0 ρ 0 ⎠ , a  = ⎝ 0 1 0 ⎠ , 0 0 1 0 0 1

(12.2)

(12.3)

where ρ = e2iπ/N . Then, all elements of Σ(3N 3 ) can be expressed as follows: ⎛ ⎞ ⎛  ⎞ ⎛ ⎞ 0 ρn 0 0 0 ρm ρ 0 0 ⎝0 ⎝ 0 ρm 0 ⎠ , ⎝ ρ 0 0 ρm ⎠ , 0 ⎠ . (12.4) ρ 0 0 0 0 ρn 0 ρn 0 For the case N = 2, the element aa  a  commutes with all the elements. In addition, when we define a˜ = aa  and a˜  = a  a  , we find the closed algebra among a, ˜ a˜  , H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_12, © Springer-Verlag Berlin Heidelberg 2012

109

110

12

Σ(3N 3 )

and b, which corresponds to Δ(12). Since the element aa  a  is not contained in this closed algebra, this group is isomorphic to Z2 × Δ(12). The situation for N = 3 is different. Once again, the element aa  a  commutes with all the elements. When we define a˜ = a 2 a  and a˜  = a  a 2 , the closed algebra among a, ˜ a˜  , and b corresponds to Δ(27). On the other hand, the element aa  a  can be written aa  a  = a˜ 2 a˜  in this case. Thus, aa  a  is one of elements of Δ(27). That is, the group Σ(81) is not Z3 × Δ(27), but isomorphic to (Z3 × Z3 × Z3 )  Z3 . Similarly, for a generic value of N , the element aa  a  commutes with all the elements. When we define a˜ = a N −1 a  and a˜  = a  a N −1 , the closed algebra generated by a, ˜ a˜  , and b corresponds to Δ(3N 2 ). For the case N/3 = integer, the element aa  a  is not included in Δ(3N 2 ). Thus, we find that this group is isomorphic to ZN × Δ(3N 2 ). On the other hand, when N/3 is an integer, the element (aa  a  )N k/3 with k = 0, 1, 2, is included in Δ(3N 2 ). That is, the group Σ(3N 3 ) cannot be ZN × Δ(3N 2 ).

12.1.1 Conjugacy Classes Here we summarize the conjugacy classes of Σ(3N 3 ): C1 :

{e},  k k k  a a a , k = 1, . . . , N − 1,   m n m n  n  m  (,m,n) ,  = m = n, : a a a ,a a a ,a a a C3   m p−−m  (p) ba a a |, m = 0, . . . , N − 1 , p = 0, . . . , N − 1, CN 2 :  2  m p −−m  (p) CN 2 : b a a a |, m = 0, . . . , N − 1 , p = 0, . . . , N − 1, (12.5) and the order h of the elements in each conjugacy class, viz., (k) C1 :

C1 :

h = 1,

(k) C1 :

h = N/ gcd(N, k),

(,m,n) C3 :

h = N/ gcd(N, , m, n),

(p) CN 2 : (p) CN 2 :

h = 3N/ gcd(N, p),

(12.6)

h = 3N/ gcd(N, p). (p)

(p)

For example, the conjugacy classes CN 2 and CN 2 are obtained with the help of the following relations:

12.1

Generic Aspects

111

  a p a q a r ba  a m a n a −p a −q a −r = ba  a m a n a q−p a r−q a p−r ,  ba p a q a r ba  a m a n a −p a −q a −r b−1 = ba n+p−r a +q−p a m+r−q ,  b2 a p a q a r ba  a m a n a −p a −q a −r b−2 = ba m+r−q a n+p−r a +q−p ,

(12.7)

where we have used a p a q a r b = ba q a r a p , a p a q a r b2 = b2 a r a p a q . Note that the sum of the factors of a, a  , and a  is the same. Since the numbers of each of (p) (p) () (,m,n) the classes C1 , C3 , CN 2 , and CN 2 are (N − 1), (N 3 − N )/3, N , and N , respectively, the total number of conjugacy classes is  1  1 + (N − 1) + N 3 − N /3 + N + N = N N 2 + 8 . 3

(12.8)

The number of irreducible representations can be determined by using the relations m1 + 4m2 + 9m3 + · · · = 3N 3 ,

1  m1 + m2 + m3 + · · · = N N 2 + 8 . (12.9) 3

Their solutions are (m1 , m2 , m3 ) = (3N, 0, N (N 2 − 1)/3). Hence the group Σ(3N 3 ) has N(N 2 + 8)/3 conjugacy classes, 3N singlets, and N (N 2 − 1)/3 triplets.

12.1.2 Characters and Representations The number of singlets is 3N . We denote them by 1k, with k = 0, 1, 2, and  = 0, . . . , N − 1. For the representations of singlets, all operators a, a  , a  , and b are mutually commuting. Then, from the algebraic relations it is found that the characters for a, a  , and a  must be the same. Thus we can represent the 3N singlets by χ1k, (a) = χ1k, (a  ) = χ1k, (a  ) = ρ  and χ1k, (b) = ωk , as shown in Table 12.1. The number of triplets is N (N 2 − 1)/3. We represent a, a  , a  , and b by ⎛ m ⎛ n ⎞ ⎞ ⎞ ⎛  ρ ρ 0 0 0 0 0 0 ρ a = ⎝ 0 ρ n 0 ⎠ , a  = ⎝ 0 ρ  0 ⎠ , a = ⎝ 0 ρm 0 ⎠ , n  0 0 ρ 0 0 ρm 0 0 ρ (12.10) and

on the triplet 3[][m][n] ,

⎛ ⎞ 0 1 0 b = ⎝0 0 1⎠ , 1 0 0 ⎞ x 3[][m][n] = ⎝xm ⎠ , xn

(12.11)

⎛

(12.12)

112

12

Σ(3N 3 )

Table 12.1 Characters of Σ(3N 3 ) h

χ1,m

χ3[][m][n]

1

1

3

(p)

N/ gcd(N, p)

ρ 3pm

3ρ p(+m+n)

(p,q,r)

N/ gcd(N, p, q, r)

ρ (p+q+r)m

ρ p+qm+rn + ρ pm+qn+r + ρ pn+q+rm

3N/ gcd(N, p)

ω ρ pm

0

3N/ gcd(N, p)

ω2 ρ pm

0

C1 C1 C3

(p) CN 2 (p) CN 2

with [][m][n] = (, m, n), (m, n, ), or (n, , m). These characters are shown in Table 12.1. The subscripts of the components give the ZN charge.

12.1.3 Tensor Products The tensor products between two triplets are given by ⎛ ⎞ ⎛ ⎞ x y ⎝xm ⎠ ⊗ ⎝ym ⎠ xn 3 yn 3    [][m][n] [ ][m ][n ] ⎞ ⎞ ⎛ ⎛ x y xm yn = ⎝xm ym ⎠ ⊕ ⎝ xn y ⎠ xn yn 3  x  y m 3  [+ ][m+m ][n+n ] [m+n ][n+ ][+m ] ⎞ ⎛ xn ym ⊕ ⎝ x yn ⎠ . xm y 3   

(12.13)

[n+m ][+n ][m+ ]

If all the subscripts become equal, the triplet can be decomposed into singlets as  (xa , xb , xc )3[k][k][k] = (xa + xb + xc )10,k + xa + ω2 xb + ωxc 1 1,k  + xa + ωxb + ω2 xc 1 . 2,k

The tensor products between singlets and triplets are ⎛ ⎞ ⎞ ⎛ x y x  ⎝xm ⎠ ⊗ (y)1k ,k = ⎝ ωk xm y ⎠  xn 3 ω2k xn y 3 [][m][n]

.

(12.14)

[+k][m+k][n+k]

The product of singlets is (x)1k, ⊗ (y)1k , = (xy)1k+k ,+ .

(12.15)

12.2

113

Σ(81)

12.2 Σ(81) We now detail the case N = 3, that is, Σ(81). It has eighty-one elements and these can be written in the form bk a m a n a  for k = 0, 1, 2, and m, n,  = 0, 1, 2, where a, a  , a  , and b satisfy a 3 = a 3 = a 3 = 1, aa  = a  a, aa  = a  a, a  a  = a  a  , b3 = 1, b2 ab = a  , b2 a  b = a, and b2 a  b = a  . These elements are classified into seventeen conjugacy classes as follows: C1 : (1)

C1 : (2)

C1 : C30) :  (0)

C3 : (1) C3 :  (1)

C3 :  (1)

C3

:

(2) C3 :  (2)

C3 :  (2)

C3

:

(0)

C9 : C9(1) : (2)

C9 : 

(0) C9 :  (1)

C9 :  (2)

C9 :

{e}, h = 1,     h = 3, aa a ,    2  , h = 3, aa a  1 2  1 2  1 2  h = 3, aa a , a a a , a a a ,  1  2 1 2 1  2  h = 3, a a a , a aa , a a a ,   1  1   1  h = 3, aa a , a aa , a a a ,  2 2  2 2  2 2  h = 3, aa a , a a a , a a a ,  1 1 2 1 1 2 1 1 2  h = 3, a a a ,a a a ,a a a ,   2  2   2  h = 3, aa a , a aa , a a a ,  1 1  1 1  1 1  h = 3, aa a , a a a , a a a ,  1 2 2 1 2 2 1 2 2  h = 3, a a a ,a a a ,a a a ,  p q −p−q  ba a a |p, q = 0, . . . , N − 1 , h = 3,  p q 1−p−q  ba a a |p, q = 0, . . . , N − 1 , h = 9,  p q 2−p−q  ba a a |p, q = 0, . . . , N − 1 , h = 9,  2 p q −p−q  b a a a |p, q = 0, . . . , N − 1 , h = 3,  2 p q 1−p−q  b a a a |p, q = 0, . . . , N − 1 , h = 9,  2 p q 2−p−q  b a a a |p, q = 0, . . . , N − 1 , h = 9,

(12.16)

where h denotes the order of each element in the given conjugacy class. The relations (2.18) and (2.19) for Σ(81) give m1 + 22 m2 + 32 m3 + · · · = 81,

(12.17)

m1 + m2 + m3 + · · · = 17.

(12.18)

114

12

Σ(3N 3 )

Table 12.2 Characters of Σ(81) for the 9 one-dimensional representations h

χ10,0

χ11,0

χ12,0

χ10,1

χ11,1

χ12,1

χ10,2

χ11,2

χ12,2

C1

1

1

1

1

1

1

1

1

1

1

(1) C1 C1(2) (0) C3  C3(0) (1) C3  C3(1)  C3 (1) (2) C3  C3(2)  (2) C3 C9(0) (1) C9 C9(2) C9(0) (1) C9 C9(2)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

3

1

1

1

1

1

1

1

1

1

3

1

1

1

1

1

1

1

1

1

3

1

1

1

ω

ω

ω

ω2

ω2

ω2

3

1

1

1

ω

ω

ω

ω2

ω2

ω2

3

1

1

1

ω

ω

ω

ω2

ω2

ω2

3

1

1

1

ω2

ω2

ω2

ω

ω

ω

3

1

1

1

ω2

ω2

ω2

ω

ω

ω

3

1

1

1

ω2

ω2

ω2

ω

ω

ω

3

1

ω

ω2

1

ω

ω2

1

ω

ω2

9

1

ω

ω2

ω

ω2

1

ω2

1

ω

9

1

ω

ω2

ω2

1

ω

ω

ω2

1

3

1

ω2

ω

1

ω2

ω

1

ω2

ω

9

1

ω2

ω

ω

1

ω2

ω2

ω

1

1

ω2

ω

ω2

ω

1

ω

1

ω2

9

We thus find (m1 , m3 ) = (9, 8), whence there are nine singlets 1k, with k,  = 0, 1, 2, and eight triplets, for which we use the notation 3A = 3[1][0][0] ,

3B = 3[0][2][2] ,

3C = 3[2][1][1] ,

3D = 3[2][0][1] ,

3¯ A = 3[2][0][0] ,

3¯ B = 3[0][1][1] ,

3¯ C = 3[1][2][2] ,

3¯ D = 3[1][0][2] .

The characters are given in Tables 12.2 and 12.3. On all the triplets, the generator b is represented by ⎛ ⎞ 0 1 0 b = ⎝0 0 1⎠. 1 0 0 The generators a, a  , and a  are represented on each triplet as follows: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ω 0 0 1 0 0 1 0 0 a = ⎝ 0 1 0⎠, a = ⎝ 0 1 0 ⎠ , a  = ⎝ 0 ω 0 ⎠ , 0 0 1 0 0 ω 0 0 1

(12.19)

(12.20)

12.2

115

Σ(81)

Table 12.3 Characters of Σ(81) for the 8 three-dimensional representations Class

n

h

χ 3A

χ3¯ A

χ3 B

χ3¯ B

χ 3C

χ3¯ C

χ3 D

χ3¯ D

C1

1

1

3

3

3

3

3

3

3

3

3

3ω

3ω2

3ω

3ω2

3ω

3ω2

3

3

3

3ω2

3ω

3ω2

3ω

3ω2

3ω

3

3

0

3ω2

3ω

3ω

3ω2

0

0

0

0

0

0

0

0

0

0

0

0

(1) C1 (2) C1 (0) C3  (0) C3 C3(1)  (1) C3  C3 (1) (2) C3  C3(2)  (2) C3 (0) C9 (1) C9 (2) C9 C9(0) (1) C9 C9(2)

on 3A ,

1 1 3

3

0

0

0

0

0

3

3

0

3

3

3

3

3

3

3

3

3

3

√ −i 3 √ −i 3ω2 √ −i 3ω √ i 3 √ i 3ω √ i 3ω2

0 √ i 3 √ i 3ω √ i 3ω2 √ −i 3 √ −i 3ω2 √ −i 3ω

0

3

0 √ i 3ω2 √ i 3 √ i 3ω √ −i 3ω √ −i 3 √ −i 3ω2

0

3

√ −i 3ω √ −i 3 √ −i 3ω2 √ 2 i 3ω √ i 3 √ i 3ω

−i 3ω2 √ −i 3ω √ −i 3 √ i 3ω √ i 3ω2 √ i 3

0 √ i 3ω √ i 3ω2 √ i 3 √ −i 3ω2 √ −i 3ω √ −i 3

9

3

0

0

0

0

0

0

0

0

9

9

0

0

0

0

0

0

0

0

9

9

0

0

0

0

0

0

0

0

9

3

0

0

0

0

0

0

0

0

9

9

0

0

0

0

0

0

0

0

9

9

0

0

0

0

0

0

0

0

⎛

⎞ 0 0 ⎠, ω2

1 0 a = ⎝ 0 ω2 0 0

on 3B ,

⎛

ω2 a=⎝ 0 0

on 3C , and ⎛ 2 ω a=⎝ 0 0

0 ω 0

⎞ 0 0 ⎠, ω

⎞ 0 0 1 0 ⎠, 0 ω

⎛

ω2  a =⎝ 0 0

0 ω2 0

⎞ 0 0⎠, 1

0 ω 0

⎞ 0 0 ⎠, ω2

1 0 a = ⎝ 0 ω 0 0

⎞ 0 0 ⎠, ω2

⎛

ω a = ⎝ 0 0

⎛

√

⎛

ω2  a =⎝ 0 0

⎛

ω a  = ⎝ 0 0

⎛

ω a  = ⎝ 0 0

⎞ 0 0 1 0 ⎠, 0 ω2 (12.21)

0 ω2 0

⎞ 0 0 ⎠, ω (12.22)

0 ω2 0

⎞ 0 0⎠, 1 (12.23)

116

12

Σ(3N 3 )

on 3D . The representations of a, a  , and a  on 3¯ A , 3¯ B , 3¯ C , and 3¯ D are obtained as complex conjugates of the representations on 3A , 3B , 3C , and 3D , respectively. On the other hand, for the singlet 1k, , these generators are represented by b = ωk ,

a = a  = a  = ω .

The tensor products between triplets are: ⎛

⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ x1 y1 x1 y1 x2 y3 x3 y2 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x3 y1 ⎠ ⊕ ⎝ x1 y3 ⎠ , x3 3 y3 3 x3 y3 3¯ x1 y2 3¯ x2 y1 3¯ A A A B B ⎛ ⎞ ⎛ ⎞ y1 x1

 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = x1 y1 + ω2k x2 y2 + ωk x3 y3 1 k,0 x3 3 y3 3¯ k=0,1,2 A A ⎛ ⎛ ⎞ ⎞ x2 y 3 x3 y2 ⊕⎝ x 3 y 1 ⎠ ⊕ ⎝ x 1 y 3 ⎠ , x1 y2 3¯ x2 y1 3 D D ⎞ ⎞ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎛ x1 y1 x1 y1 x3 y2 x2 y3 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x1 y3 ⎠ ⊕ ⎝ x3 y1 ⎠ , x3 3 y3 3 x3 y3 3¯ x2 y1 3¯ x1 y2 3¯ A B C A A ⎛ ⎞ ⎛ ⎞ y1 x1

 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = x1 y1 + ω2k x2 y2 + ωk x3 y3 1 k,1 x3 3 y3 3¯ k=0,1,2 A B ⎛ ⎛ ⎞ ⎞ x2 y 3 x3 y2 ⊕⎝ x 3 y 1 ⎠ ⊕ ⎝ x 1 y 3 ⎠ , x1 y2 3¯ x2 y1 3 D D ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎛ ⎞ y1 x1 y1 x2 y3 x3 y2 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x3 y1 ⎠ ⊕ ⎝ x1 y3 ⎠ , x3 3 y3 3 x3 y3 3¯ x1 y2 3¯ x2 y1 3¯ A C B C C ⎛ ⎞ ⎛ ⎞ x1 y1

 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = x1 y1 + ω2k x2 y2 + ωk x3 y3 1 k,2 x3 3 y3 3¯ k=0,1,2 A C ⎛ ⎛ ⎞ ⎞ x2 y 3 x3 y2 ⊕⎝ x 3 y 1 ⎠ ⊕ ⎝ x 1 y 3 ⎠ , x1 y2 3¯ x2 y1 3 D

⎛

A

⎞

⎛

D

⎞

(12.25)

(12.26)

(12.27)

(12.28)

(12.29)

D

⎞ ⎛ y1 x3 y 3 x3 y2 x3 y1 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x1 y1 ⎠ ⊕ ⎝ x1 y3 ⎠ ⊕ ⎝ x1 y2 ⎠ , (12.30) x3 3 y3 3 x2 y2 3 x2 y1 3 x2 y3 3 ⎛

⎞

(12.24)

⎛

A

⎞

B

C

12.2

117

Σ(81)

⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ y1 x2 y 1 x2 y2 x2 y3 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x3 y2 ⎠ ⊕ ⎝ x3 y3 ⎠ ⊕ ⎝ x3 y1 ⎠ , (12.31) x3 3 y3 3¯ x1 y3 3 x1 y1 3 x1 y2 3 ⎛

A

⎛

⎞

⎛

⎞

⎛

⎞

⎛

⎞

D

⎛

⎞

⎛

⎞

⎛

⎞

⎛

⎞

A

⎛

⎞

⎛

⎞

B

⎛

⎞

⎛

⎞

C

⎛

⎞

⎛

⎞

x1 y1 x1 y1 x3 y2 x2 y3 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x1 y3 ⎠ ⊕ ⎝ x3 y1 ⎠ , (12.32) x3 3¯ y3 3¯ x3 y3 3 x2 y1 3 x1 y2 3 A B C A A ⎛ ⎞ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ y1 x1 y1 x3 y2 x2 y3 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x1 y3 ⎠ ⊕ ⎝ x3 y1 ⎠ , (12.33) x3 3¯ y3 3¯ x3 y3 3 x2 y1 3 x1 y2 3 A

C

B

C

C

x1 y1 x1 y 1 x3 y2 x2 y3 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x1 y3 ⎠ ⊕ ⎝ x3 y1 ⎠ , (12.34) x3 3 y3 3 x3 y3 3¯ x2 y1 3¯ x1 y2 3¯ B C C B B ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎛ ⎞ y1 x1 y 1 x3 y2 x2 y3 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x1 y3 ⎠ ⊕ ⎝ x3 y1 ⎠ , (12.35) x3 3 y3 3 x3 y3 3¯ x2 y1 3¯ x1 y2 3¯ B C A B B ⎛ ⎞ ⎞ ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ ⎛ x1 y1 x3 y 3 x3 y2 x3 y1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x1 y1 ⎠ ⊕ ⎝ x1 y3 ⎠ ⊕ ⎝ x1 y2 ⎠ , (12.36) x3 3 y3 3 x2 y2 3 x2 y1 3 x2 y3 3 B

D

B

C

A

y1 x1

 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = x1 y1 + ω2k x2 y2 + ωk x3 y3 1 k,0 x3 3¯ y3 3 k=0,1,2 B B ⎞ ⎞ ⎛ ⎛ x1 y 2 x2 y1 ⊕⎝ x 2 y 3 ⎠ ⊕ ⎝ x 3 y 2 ⎠ , (12.37) x3 y1 3 x1 y3 3¯ D D ⎛ ⎞ ⎛ ⎞ y1 x1

 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = x1 y1 + ω2k x2 y2 + ωk x3 y3 1 k,2 x3 3¯ y3 3 k=0,1,2 B C ⎛ ⎛ ⎞ ⎞ x2 y 3 x1 y3 ⊕⎝ x 3 y 1 ⎠ ⊕ ⎝ x 2 y 1 ⎠ , (12.38) x1 y2 3 x3 y2 3¯ D D ⎞ ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ ⎛ ⎛ ⎞ y1 x2 y2 x2 y1 x2 y3 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x3 y3 ⎠ ⊕ ⎝ x3 y2 ⎠ ⊕ ⎝ x3 y1 ⎠ , (12.39) x3 3¯ y3 3 x1 y1 3¯ x1 y3 3¯ x1 y2 3¯ B

D

⎛

⎞

C

⎛

⎞

B

⎛

⎞

A

x1 y1 x1 y1 x3 y2 x2 y3 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x1 y3 ⎠ ⊕ ⎝ x3 y1 ⎠ , (12.40) x3 3 y3 3 x3 y3 3¯ x2 y1 3¯ x1 y2 3¯ C

C

C

A

A

118

12

Σ(3N 3 )

⎛ ⎞ ⎞ y1 x1

 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = x1 y1 + ω2k x2 y2 + ωk x3 y3 1 k,0 x3 3¯ y3 3 k=0,1,2 C C ⎞ ⎞ ⎛ ⎛ x1 y 2 x2 y1 ⊕⎝ x 2 y 3 ⎠ ⊕ ⎝ x 3 y 2 ⎠ , (12.41) x3 y1 3 x1 y3 3¯ D D ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎛ ⎞ y1 x2 y2 x2 y1 x2 y3 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x3 y3 ⎠ ⊕ ⎝ x3 y2 ⎠ ⊕ ⎝ x3 y1 ⎠ , (12.42) x3 3¯ y3 3 x1 y1 3¯ x1 y3 3¯ x1 y2 3¯ ⎛

C

A

D

C

B

⎞ ⎞ ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ ⎛ x1 y1 x1 y1 x2 y3 x3 y2 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x3 y1 ⎠ ⊕ ⎝ x1 y3 ⎠ , (12.43) x3 3 y3 3 x3 y3 3¯ x1 y2 3¯ x2 y1 3¯ D D D D D ⎛ ⎞ ⎛ ⎞ y1 x1

 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = x1 y1 + ω2k x2 y2 + ωk x3 y3 1 k,0 x3 3 y3 3¯ k=0,1,2 D D  ⊕ x2 y3 + ω2k x3 y1 + ωk x1 y2 1 ⎛

k,2

 ⊕ x3 y2 + ω2k x1 y3 + ωk x2 y1 1 . k,1

(12.44) The tensor products between singlets are 1k, ⊗ 1k  , = 1k+k  (mod 3),+

(mod 3) .

(12.45)

The tensor products between singlets and triplets are: ⎞ ⎞ ⎛ y1 xy1 = ⎝ xy2 ⎠ , (x)10,0 ⊗ ⎝ y2 ⎠ y3 3(3) xy 3 3(3) ¯ A ¯ A ⎞ ⎛ ⎞ ⎛ xy1 y1 ⎠ ⎝ ⎝ ωxy = , (x)11,0 ⊗ y2 2 ⎠ 2 y3 3(3) ω xy3 3(3) ¯ A ¯ A ⎞ ⎛ ⎞ ⎛ xy1 y1 = ⎝ ω2 xy2 ⎠ , (x)12,0 ⊗ ⎝ y2 ⎠ y3 3(3) ωxy3 3(3) ¯ ¯ ⎛

A

A

(12.46)

12.2

119

Σ(81)

⎛

⎞ y1 (x)10,1 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ A ⎛ ⎞ y1 (x)11,1 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ A ⎛ ⎞ y1 (x)12,1 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ A ⎛ ⎞ y1 (x)10,2 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ A ⎛ ⎞ y1 (x)11,2 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ A ⎛ ⎞ y1 (x)12,2 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ A ⎛ ⎞ y1 (x)10,0 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ B ⎛ ⎞ y1 (x)11,0 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ B ⎛ ⎞ y1 (x)12,0 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ B ⎛ ⎞ y1 (x)10,1 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ B ⎛ ⎞ y1 ⎝ y (x)11,1 ⊗ 2⎠ y3 3(3) ¯ B ⎛ ⎞ y1 (x)12,1 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ B

⎛

⎞ xy1 = ⎝ xy2 ⎠ , xy3 3 ,(3¯ ) C B ⎛ ⎞ xy1 = ⎝ ωxy2 ⎠ , ω2 xy3 3 ,(3¯ ) C B ⎞ ⎛ xy1 = ⎝ ω2 xy2 ⎠ , ωxy3 3 ,(3¯ ) C B ⎛ ⎞ xy1 = ⎝ xy2 ⎠ , xy3 3 ,(3¯ ) B C ⎛ ⎞ xy1 = ⎝ ωxy2 ⎠ , ω2 xy3 3 ,(3¯ ) B C ⎞ ⎛ xy1 = ⎝ ω2 xy2 ⎠ , ωxy3 3 ,(3¯ ) B C ⎛ ⎞ xy1 = ⎝ xy2 ⎠ , xy3 3(3) ¯ B ⎛ ⎞ xy1 = ⎝ ωxy2 ⎠ , ω2 xy3 3(3) ¯ B ⎞ ⎛ xy1 = ⎝ ω2 xy2 ⎠ , ωxy3 3(3) ¯ B ⎛ ⎞ xy1 = ⎝ xy2 ⎠ , xy3 3 ,(3¯ ) A C ⎛ ⎞ xy1 = ⎝ ωxy2 ⎠ , ω2 xy3 3 ,(3¯ ) A C ⎞ ⎛ xy1 = ⎝ ω2 xy2 ⎠ , ωxy3 3 ,(3¯ ) A

C

(12.47)

(12.48)

(12.49)

(12.50)

120

12

⎛

⎞ y1 (x)10,2 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ B ⎛ ⎞ y1 (x)11,2 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ B ⎛ ⎞ y1 (x)12,2 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ B ⎛ ⎞ y1 (x)10,0 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ C ⎛ ⎞ y1 (x)11,0 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ C ⎛ ⎞ y1 (x)12,0 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ C ⎛ ⎞ y1 (x)10,1 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ C ⎛ ⎞ y1 (x)11,1 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ C ⎛ ⎞ y1 (x)12,1 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ C ⎛ ⎞ y1 (x)10,2 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ C ⎛ ⎞ y1 ⎝ y (x)11,2 ⊗ 2⎠ y3 3(3) ¯ C ⎛ ⎞ y1 (x)12,2 ⊗ ⎝ y2 ⎠ y3 3(3) ¯ C

Σ(3N 3 )

⎛

⎞ xy1 = ⎝ xy2 ⎠ , xy3 3 ,(3¯ ) C A ⎛ ⎞ xy1 = ⎝ ωxy2 ⎠ , ω2 xy3 3 ,(3¯ ) C A ⎞ ⎛ xy1 = ⎝ ω2 xy2 ⎠ , ωxy3 3 ,(3¯ ) C A ⎛ ⎞ xy1 = ⎝ xy2 ⎠ , xy3 3(3) ¯ C ⎛ ⎞ xy1 = ⎝ ωxy2 ⎠ , ω2 xy3 3(3) ¯ C ⎞ ⎛ xy1 ⎝ = ω2 xy2 ⎠ , ωxy3 3(3) ¯ C ⎛ ⎞ xy1 = ⎝ xy2 ⎠ , xy3 3 ,(3¯ ) B A ⎛ ⎞ xy1 = ⎝ ωxy2 ⎠ , ω2 xy3 3 ,(3¯ ) B A ⎞ ⎛ xy1 = ⎝ ω2 xy2 ⎠ , ωxy3 3 ,(3¯ ) B A ⎛ ⎞ xy1 = ⎝ xy2 ⎠ , xy3 3 ,(3¯ ) A B ⎛ ⎞ xy1 = ⎝ ωxy2 ⎠ , ω2 xy3 3 ,(3¯ ) A B ⎞ ⎛ xy1 = ⎝ ω2 xy2 ⎠ , ωxy3 3 ,(3¯ ) A

B

(12.51)

(12.52)

(12.53)

(12.54)

References

121

⎛

⎛ ⎞ ⎞ y1 xy1 (x)1k, ⊗ ⎝ y2 ⎠ = ⎝ xy2 ⎠ , y3 3(3) xy3 3(3) ¯ ¯ D

where k,  = 0, 1, 2.

References 1. Ishimori, H., Kobayashi, T.: arXiv:1201.3429 [hep-ph]

D

(12.55)

Chapter 13

Δ(6N 2 )

In this chapter, we investigate the discrete group Δ(6N 2 ), which is isomorphic to  )  S (see also [1]). Let us denote the generators of Z and Z  by a (ZN × ZN 3 N N  and a , respectively. We denote the S3 generators by b and c, where b and c are the Z3 and Z2 generators of S3 , respectively. These satisfy a N = a  = b3 = c2 = (bc)2 = e, aa  = a  a,  −1 bab−1 = a −1 a  , ba  b−1 = a,  −1 cac−1 = a  , ca  c−1 = a −1 . N

(13.1)

Using these, all elements of Δ(6N 2 ) can be expressed in the form g = bk c a m a n ,

(13.2)

for k = 0, 1, 2,  = 0, 1, and m, n = 0, 1, 2, . . . , N − 1. Note that the group Δ(6N 2 ) includes as a subgroup Δ(3N 2 ), whose elements can be written bk a m a n . Thus, some group-theoretical aspects of Δ(6N 2 ) can be derived from those of Δ(3N 2 ).

13.1 Δ(6N 2 ) with N/3 = Integer 13.1.1 Conjugacy Classes With a view to identifying the conjugacy classes, we note that aba −1 = ba −1 a  ,

a  ba −1 = ba −1 a −2 ,

aca −1 = ca −1 a −1 ,

a  ca −1 = ca −1 a −1 ,

cbc−1

bcb−1

= b2 ,

(13.3)

= b2 c.

H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_13, © Springer-Verlag Berlin Heidelberg 2012

123

124

13

Δ(6N 2 )

Using these relations, one can obtain the conjugacy classes of Δ(6N 2 ). Indeed, these relations are nothing but those in Δ(3N 2 ), except for the relations involving c. Hence, the conjugacy classes of Δ(3N 2 ) are useful to obtain those of Δ(6N 2 ). First, we consider the elements a  a m . As shown in Chap. 10, the element a  a m is conjugate to a −+m a − and a −m a −m for any group Δ(3N 2 ) with N/3 = integer. These elements must therefore be conjugate to each other in Δ(6N 2 ), too. In addition, it is found that ca  a m c−1 = a −m a − ,

ca −+m a − c−1 = a  a −m ,

ca −m a −m c−1 = a −+m a m .

(13.4)

Thus, the following elements, a  a m , a −m a − ,

a −+m a − ,

a −m a −m ,

a  a −m ,

a −+m a m ,

(13.5)

are conjugate to each other in Δ(6N 2 ). However, the elements, a −m a − , a  a −m , and a −+m a m are the same as a  a m when  and m satisfy the conditions  + m = 0 mod(N ),

2 − m = 0 mod(N ),

 − 2m = 0 mod(N ), (13.6)

respectively. Under these conditions, the above conjugate elements in (13.5) reduce to the three elements a  a − , a −2 a − , and a  a 2 . As a result, the elements a  a m are classified into the following conjugacy classes:   (k) (13.7) C3 = a k a −k , a −2k a −k , a k a 2k , k = 1, 2, . . . , N − 1,   (,m) = a  a m , a m− a − , a −m a −m , a −m a − , a m− a m , a  a −m , (13.8) C6 (,m) run from 0 to N − 1, but they for N/3 = integer. The parameters  and m of C6 do not satisfy the conditions (13.6). The numbers of each of the conjugacy classes (k) (,m) C3 and C6 are (N − 1) and (N 2 − 3N + 2)/6, respectively. Similarly, we can obtain conjugacy classes containing ba  a m and b2 a  a m . As shown in Chap. 10, all the elements ba  a m with , m = 0, . . . , N − 1, belong to the same conjugacy classes in the group Δ(3N 2 ). In addition, all the elements b2 a  a m with , m = 0, . . . , N − 1, also belong to the same conjugacy classes in Δ(3N 2 ). Furthermore, we obtain

cba  a m c−1 = b2 a −m a − .

(13.9)

Thus, all the elements bk a  a m , for k = 1, 2, and , m = 0, . . . , N − 1, belong to the same conjugacy class in Δ(6N 2 ). Finally, we consider the conjugate elements involving c, which are not included in Δ(3N 2 ). Here we find that  p q   m  p q −1 a a ca a a a = ca −p−q a m−p−q = ca k+n a n , (13.10)

13.1

Δ(6N 2 ) with N/3 = Integer

125



  −1 ba p a q ca  a m ba p a q = b2 ca −+m a −+p+q = b2 ca −k a −k−n ,  2 p q   m  2 p q −1 = bca −m+p+q a −m = bca −n a k , b a a ca a b a a

(13.11) (13.12)

where k =  − m and n = m − p − q. As a result, all the elements ca k+n a n ,

b2 ca −k a −k−n ,

bca −n a k ,

(13.13)

with n = 0, 1, . . . , N − 1, belong to the same conjugacy class. Here, we summarize the conjugacy classes of Δ(6N 2 ). For N/3 = integer, Δ(6N 2 ) has the following conjugacy classes: C1 :

{e},  k −k −2k −k k 2k  , k = 1, 2, . . . , N − 1, a a ,a a ,a a   m m− − −m −m −m − m− m  −m  (,m) , a ,a a ,a a ,a a ,a a C6 : a a , a (13.14) , m = 0, 1, . . . , N − 1,   k  m  b a a  k = 1, 2, , m = 0, 1, . . . , N − 1 , C2N 2 :    () C3N : ca +n a n , b2 ca − a −−n , bca −n a   n = 0, 1, . . . , N − 1 ,  = 0, 1, . . . , N − 1, (k) C3 :

where  and m in C6(,m) do not satisfy the conditions (13.6). The order h of each element in the given conjugacy class is: C1 :

h = 1,

(k) C3 :

h = N/ gcd(N, k),

(,m) C6 :

h = N/ gcd(N, , m),

C2N 2 :

h = 3,

()

C3N :

(13.15)

h = 2N/ gcd(N, ).

() Since the numbers of each of the conjugacy classes C3(k) , C6(,m) , and C3N are N − 1, (N 2 −3N +2)/6, and N , respectively, the total number of conjugacy classes is equal to (N 2 + 9N + 8)/6. The relations (2.18) and (2.19) for Δ(6n2 ) with N/3 = integer give

m1 + 22 m2 + 32 m3 + · · · = 6N 2 , m1 + m2 + m3 + · · · =

N 2 + 9N + 8 . 6

(13.16) (13.17)

These have the solution (m1 , m2 , m3 , m6 ) = (2, 1, 2(N − 1), (N 2 − 3N + 2)/6).

126

13

Δ(6N 2 )

13.1.2 Characters and Representations The group Δ(6N 2 ) with N/3 = integer has two singlets and one doublet. These are nothing but the irreducible representations of S3 . Thus, on these representations, the  generators a and a  are identity matrices. Since c2 = e, the characters ZN and ZN of the two singlets have two possible values χ1k (c) = (−1)k with k = 0, 1, and they correspond to two singlets 1k . Note that χ1k (a) = χ1k (a  ) = χ1k (b) = 1. Similarly to S3 , the characters of the doublet 2 are obtained as  (k)  = 0, χ2 C3N

χ2 (C2N 2 ) = −1,

(13.18)

together with χ2 (a) = χ2 (a  ) = 2. The two-dimensional representations are, e.g.,  b(2) =

ω 0

 0 , ω2

c(2) =

 0 1

 1 , 0

a(2) = a  (2) =



 1 0 . (13.19) 0 1

Next, we consider the sextet representations. We can obtain (6 × 6) matrix representations for the generic sextet. The group Δ(6N 2 ) is represented as follows:  b=

b1 0

 0 , b2

 c=

 0 1 , 1 0

 a=

a1 0

 0 , a2

a =

  a1 0

 0 , a2 (13.20)

where ⎛ 0 b1 = ⎝0 1

⎞ ⎛ ⎞ 0 0 0 1 1⎠ , b2 = ⎝1 0 0⎠ , 0 0 1 0 ⎛ l ⎞ 0 0 ρ −1 a1 = a  2 = ⎝ 0 ρ k 0 ⎠, 0 0 ρ −l−k ⎛ l+k ⎞ ρ 0 0 −1 a2 = a  1 = ⎝ 0 0 ⎠, ρ −l 0 0 ρ −k 1 0 0

(13.21)

(13.22)

on the sextet 6[[k],[]] with (k, ) = (0, 0), where [[k], []] is defined by1

 [k], [] = (k, ), (−k − , k), (, −k − ), (−, −k), (k + , −), or(−k, k + ). (13.23)

1 The

 notation [[k], []] corresponds to (k, ) in [1].

13.1

Δ(6N 2 ) with N/3 = Integer

127

We also denote the vector 6[[k],[]] by ⎞ x,−k− ⎜ xk, ⎟ ⎟ ⎜ ⎜x−k−,k ⎟ ⎟, ⎜ 6[[k],[]] = ⎜ ⎟ ⎜xk+,− ⎟ ⎝ x−,−k ⎠ x−k,k+ ⎛

(13.24)

 charges, for k,  = 0, 1, . . . , N − 1, where k and  correspond to the ZN and ZN respectively. In certain cases, the above representation becomes reducible. For N/3 = integer, 6[[k],[]] is reducible if k +  = 0 (mod N ), k = 0,  = 0, and  = 0, k = 0, and then the number of irreducible sextet representations is [N 2 − N − 2(N − 1)]/6. On the other hand, we can diagonalize the above reducible 6D representations so as to obtain 2(N − 1) irreducible triplets 31k and 32k , with k = 1, . . . , N − 1, where the generators are represented by

⎛

0 1 b(31k ) = ⎝0 0 1 0 ⎛ k ρ 0 a(31k ) = ⎝ 0 ρ −k 0 0

⎞ 0 1⎠ , 0 ⎞ 0 0⎠ , 1

⎛ 0 c(31k ) = ⎝0 1 ⎛ 1 a  (31k ) = ⎝0 0

⎞ 0 1 1 0⎠ , 0 0 0 ρk 0

⎞ 0 0 ⎠,

(13.25)

ρ −k

b(32k ) = b(31k ),

c(32k ) = −c(31k ),

a(32k ) = a(31k ),

a  (32k ) = a  (31k ).

(13.26)

We also denote the vectors 31k and 32k by ⎞ xk,0 31k = ⎝x−k,k ⎠ , x0,−k ⎛

for k = 1, . . . , N − 1. The characters are shown in Table 13.1.

⎞ xk,0 32k = ⎝x−k,k ⎠ , x0,−k ⎛

(13.27)

128

13

Δ(6N 2 )

Table 13.1 Characters of Δ(6N 2 ) for N/3 = integer h

χ 1r

χ2 χ31k

χ32k

χ6[[k],[]]

C1

1

1

2 3

3

6

C3(m)

N/ gcd(N, m)

1

2 ρ −2mk + 2ρ mk

ρ −2mk + 2ρ mk

2ρ m(k−) + 2ρ −m(2k+) + 2ρ m(k+2)

2 ρ mk + ρ −nk + ρ (−m+n)k

ρ mk + ρ −nk + ρ (−m+n)k

ρ mk+n + ρ (−m+n)k−m + ρ −nk+(m−n) + ρ −nk−m + ρ mk+(−m+n) + ρ (−m+n)k+n

0

0

−ρ −mk

0

C6(m,n) N/ gcd(N, m, n) 1

C2N 2

3

1

(m) C3N

2N/ gcd(N, m)

(−1)r

−1 0 0 ρ −mk

13.1.3 Tensor Products

 charges, tensor products of sextets 6 can be obtained as Because of their ZN × ZN follows:

⎞ x,−k− ⎜ xk, ⎟ ⎟ ⎜ ⎜x−k−,k ⎟ ⎟ ⎜ ⎜xk+,− ⎟ ⎟ ⎜ ⎝ x−,−k ⎠ x−k,k+ 6 ⎛

[[k],[]]

⎞ y ,−k  − ⎜ yk , ⎟ ⎟ ⎜ ⎜y−k  − ,k  ⎟ ⎟ ⎜ ⊗⎜ ⎟ ⎜yk + ,− ⎟ ⎝ y− ,−k  ⎠ y−k  ,k  + 6 ⎛

[[k  ],[ ]]

⎞ x,−k− y ,−k  − ⎟ ⎜ xk, yk , ⎟ ⎜ ⎜x−k−,k y−k  − ,k  ⎟ ⎟ =⎜ ⎜xk+,− yk  + ,− ⎟ ⎟ ⎜ ⎝ x−,−k y− ,−k  ⎠ x−k,k+ y−k  ,k  + 6 ⎛

[[k+k  ],[+ ]]

⎛

⎞ x,−k− yk  , ⎜ xk, y−k  − ,k  ⎟ ⎟ ⎜ ⎜x−k−,k y ,−k  − ⎟ ⎟ ⊕⎜ ⎜ xk+,− y− ,−k  ⎟ ⎟ ⎜ ⎝ x−,−k y−k  ,k  + ⎠ x−k,k+ yk + ,− 6 ⎛

⎞

x,−k− y−k  − ,k ⎜ xk, y ,−k  − ⎟ ⎟ ⎜ ⎜ x−k−,k yk  , ⎟ ⎟ ⎜ ⊕⎜ ⎟ ⎜xk+,− y−k  ,k  + ⎟ ⎝ x−,−k yk + ,− ⎠ x−k,k+ y− ,−k  6

[[k+ ],[−k  − ]]

[[k−k  − ],[+k  ]]

⎞ x,−k− y−k  ,k  + ⎜ xk, y− ,−k  ⎟ ⎟ ⎜ ⎜x−k−,k yk + ,− ⎟ ⎟ ⎜ ⊕⎜ ⎟ ⎜xk+,− y−k  − ,k ⎟ ⎝ x−,−k yk , ⎠ x−k,k+ y ,−k  − 6 ⎛

[[k− ],[−k  ]]

13.1

Δ(6N 2 ) with N/3 = Integer

129

⎞ x,−k− y− ,−k  ⎜ xk, yk + ,− ⎟ ⎟ ⎜ ⎜x−k−,k y−k  ,k  + ⎟ ⎟ ⎜ ⊕⎜ ⎟ ⎜ xk+,− yk  , ⎟ ⎝ x−,−k y ,−k  − ⎠ x−k,k+ y−k  − ,k 6 ⎛

[[k+k  + ],[− ]]

⎞ ⎛ x,−k− yk + ,− ⎜ xk, y−k  ,k  + ⎟ ⎟ ⎜ ⎜ x−k−,k y− ,−k  ⎟ ⎟ ⎜ ⊕⎜ ⎟ ⎜xk+,− y ,−k  − ⎟ ⎝ x−,−k y−k  − ,k  ⎠ x−k,k+ yk  , 6

.

[[k−k  ],[+k  + ]]

(13.28) Similarly, products of the sextets 6 and the triplets 31k and 32k are: ⎞ x,−k− ⎜ xk, ⎟ ⎞ ⎛ ⎟ ⎜ yk  ,0 ⎜x−k−,k ⎟ ⎟ ⎜ ⊗ ⎝y−k  ,k  ⎠ ⎜xk+,− ⎟ ⎟ ⎜ y0,−k  3  ⎝ x−,−k ⎠ 1k x−k,k+ 6 [[k],[]] ⎞ ⎞ ⎛ ⎛ x,−k− y−k  ,k x,−k− yk ,0 ⎜ xk, y0,−k  ⎟ ⎜ xk, y−k  ,k  ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ x−k−,k yk ,0 ⎟ ⎜x−k−,k y0,−k  ⎟ ⎟ ⎟ ⎜ =⎜ ⊕ ⎜xk+,− y−k  ,k ⎟ ⎜xk+,− y0,−k  ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ x−,−k yk  ,0 ⎠ ⎝ x−,−k y−k  ,k  ⎠ x−k,k+ y0,−k  6 x−k,k+ yk  ,0 6  [[k],[−k  ]] [[k−k ],[+k  ]] ⎞ ⎛ x,−k− y0,−k  ⎜ xk, yk  ,0 ⎟ ⎟ ⎜ ⎜x−k−,k y−k  ,k  ⎟ ⎟ ⎜ ⊕⎜ , ⎟ ⎜ xk+,− yk  ,0 ⎟ ⎝ x−,−k y0,−k  ⎠ x−k,k+ y−k  ,k  6  [[k+k ],[]] ⎞ ⎛ x,−k− ⎜ xk, ⎟ ⎛ ⎞ ⎟ ⎜ yk  ,0 ⎜x−k−,k ⎟ ⎟ ⎜ ⊗ ⎝y−k  ,k  ⎠ ⎜xk+,− ⎟ ⎟ ⎜ y0,−k  3  ⎝ x−,−k ⎠ 2k x−k,k+ 6 [[k],[]] ⎞ ⎞ ⎛ ⎛ x,−k− y−k  ,k  x,−k− yk ,0 ⎜ xk, y0,−k  ⎟ ⎜ xk, y−k  ,k  ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ x−k−,k yk ,0 ⎟ ⎜ x−k−,k y0,−k  ⎟ ⎟ ⎟ ⎜ ⎜ =⎜ ⊕⎜ ⎟ ⎟ ⎜−xk+,− y−k  ,k  ⎟ ⎜−xk+,− y0,−k  ⎟ ⎝ −x−,−k yk ,0 ⎠ ⎝ −x−,−k y−k  ,k ⎠ −x−k,k+ y0,−k  6 −x−k,k+ yk  ,0 6   ⎛

[[k],[−k ]]

[[k−k ],[+k  ]]

(13.29)

130

Δ(6N 2 )

13

⎞ x,−k− y0,−k  ⎟ ⎜ xk, yk ,0 ⎟ ⎜ ⎜ x−k−,k y−k  ,k  ⎟ ⎟ ⎜ ⊕⎜ ⎟ ⎜ −xk+,− yk  ,0 ⎟ ⎝ −x−,−k y0,−k  ⎠ −x−k,k+ y−k  ,k  6 ⎛

(13.30)

,

[[k+k  ],[]]

⎞ ⎛ x0,−k y−k  ,k ⎜ xk,0 y0,−k  ⎟ ⎛ ⎞ ⎟ ⎜ xk,0 yk ,0 ⎜ x−k,k yk  ,0 ⎟ ⎟ ⎜ ⎝ ⎠ = x−k,k y−k  ,k  ⊕⎜ xk,0 y−k  ,k  ⎟ ⎟ ⎜ x0,−k y0,−k  3 ⎝ x0,−k yk  ,0 ⎠ 1(k+k  ) x−k,k y0,−k  6

⎛

⎛ ⎞ ⎞ xk,0 yk ,0 ⎝x−k,k ⎠ ⊗ ⎝y−k  ,k  ⎠ x0,−k 3 y0,−k  3

1k 

1k

,

[[k],[−k  ]]

(13.31)

⎞ x0,−k y−k  ,k ⎜ xk,0 y0,−k  ⎟ ⎛ ⎞ ⎟ ⎜ xk,0 yk  ,0 ⎜ x−k,k yk  ,0 ⎟ ⎟ ⎜ ⎝ ⎠ = x−k,k y−k  ,k  ⊕⎜ −xk,0 y−k  ,k  ⎟ ⎟ ⎜ x0,−k y0,−k  3 ⎝ −x0,−k yk  ,0 ⎠ 2(k+k  ) −x−k,k y0,−k  6 ⎛

⎛ ⎞ ⎞ yk  ,0 xk,0 ⎝x−k,k ⎠ ⊗ ⎝y−k  ,k  ⎠ x0,−k 3 y0,−k  3 ⎛

2k 

1k

,

[[k],[−k  ]]

(13.32)

⎛ ⎞ ⎞ yk ,0 xk,0 ⎝x−k,k ⎠ ⊗ ⎝y−k  ,k  ⎠ x0,−k 3 y0,−k  3 ⎛

2k 

2k

⎞ ⎛ x0,−k y−k  ,k ⎜ xk,0 y0,−k  ⎟ ⎛ ⎞ ⎟ ⎜ xk,0 yk ,0 ⎜ x−k,k yk  ,0 ⎟ ⎟ = ⎝x−k,k y−k  ,k  ⎠ ⊕⎜ ⎜ xk,0 y−k  ,k  ⎟ ⎟ ⎜ x0,−k y0,−k  3 ⎝ x0,−k yk  ,0 ⎠ 1(k+k  ) x−k,k y0,−k  6

,

[[k],[−k  ]]

(13.33)

⎛

⎞ x,−k− ⎜ xk, ⎟ ⎜ ⎟ ⎜x−k−,k ⎟ ⎜ ⎟ ⎜xk+,− ⎟ ⎜ ⎟ ⎝ x−,−k ⎠ x−k,k+ 6 ⎛

⎞ x,−k− y1 ⎜ ωxk, y1 ⎟ ⎟ ⎜ 2   ⎜ω x−k−,k y1 ⎟ y1 ⎟ ⎜ ⊗ = ⎟ y2 2 ⎜ ⎜ xk+,− y2 ⎟ ⎝ ωx−,−k y2 ⎠ ω2 x−k,k+ y2 6 [[k],[]]

⎞

⎛

[[k],[]]

⎛

⎞

⎞ x,−k− y2 ⎜ ω2 xk, y2 ⎟ ⎟ ⎜ ⎜ωx−k−,k y2 ⎟ ⎟ ⎜ ⊕⎜ ⎟ ⎜ xk+,− y1 ⎟ 2 ⎝ω x−,−k y1 ⎠ ωx−k,k+ y1 6 ⎛

⎛

,

[[k],[]]

(13.34) ⎞

  xk,0 xk,0 (y1 + ω2 y2 ) xk,0 (y1 − ω2 y2 ) y 1 ⎝x−k,k ⎠ ⊗ = ⎝ ωx−k,k (y1 + y2 ) ⎠ ⊕ ⎝ ωx−k,k (y1 − y2 ) ⎠ , y2 2 x0,−k 3 x0,−k (ω2 y1 + y2 ) 3 x0,−k (ω2 y1 − y2 ) 3 1k 1k 2k (13.35)

13.2

Δ(6N 2 ) with N/3 Integer

131

⎞ ⎞ ⎛ ⎛ ⎞   xk,0 (y1 + ω2 y2 ) xk,0 (y1 − ω2 y2 ) xk,0 y 1 ⎝x−k,k ⎠ ⊗ = ⎝ ωx−k,k (y1 + y2 ) ⎠ ⊕ ⎝ ωx−k,k (y1 − y2 ) ⎠ , y2 2 x0,−k 3 x0,−k (ω2 y1 + y2 ) 3 x0,−k (ω2 y1 − y2 ) 3 2k 2k 1k (13.36)       x1 y1 x2 y2 ⊗ = (x1 y2 + x2 y1 )10 ⊕ (x1 y2 − x2 y1 )11 ⊕ . (13.37) x2 2 y2 2 x1 y1 2 ⎛

Tensor products with trivial singlets remain the same representation, while products with non-trivial singlets are: ⎞ ⎛ ⎛ ⎞ x1 y x1 ⎜ x2 y ⎟ ⎜x2 ⎟ ⎛ ⎞ ⎛ ⎞ ⎟ ⎜ ⎜ ⎟ x1 x1 y ⎜ x3 y ⎟ ⎜x3 ⎟ ⎟ ⎜ ⎜ ⎟ ⎝x2 ⎠ ⊗ (y)11 = ⎝x2 y ⎠ , ⊗ (y) = , 1 1 ⎜−x4 y ⎟ ⎜x4 ⎟ ⎟ ⎜ ⎜ ⎟ x3 3 x3 y 3 ⎝−x5 y ⎠ ⎝x5 ⎠ 1k 2k x6 6 −x6 y 6 [[k],[]]

[[k],[]]

⎛ ⎞ ⎛ ⎞ x1 y x1 ⎝x2 ⎠ ⊗ (y)11 = ⎝x2 y ⎠ , x3 y 3 x3 3 2k

(13.38)     x1 x1 y ⊗ (y)11 = . x2 2 −x2 y 2

(13.39)

1k

13.2 Δ(6N 2 ) with N/3 Integer 13.2.1 Conjugacy Classes The conjugacy classes of Δ(3N 2 ) are useful to obtain those of Δ(6N 2 ), except for the elements involving c. Conjugacy classes containing the elements a  a m are almost the same as those of Δ(6N 2 ) when N/3 = integer. One difference is that, when N/3 is an integer, there are classes with one element. Since ba  a − b−1 = a −2 a − ,

ca  a − c−1 = a  a − ,

(13.40)

and  = −2 if  = N/3 or 2N/3, there are classes {a N/3 a 2N/3 } and {a 2N/3 a N/3 }. Now consider the elements ba  a m and b2 a  a m . In Δ(3N 2 ) with N/3 integer, the elements ba p−n−3m a n with m = 0, 1, . . . , (N − 3)/3, and n = 0, 1, . . . , N − 1, belong to the same conjugacy class. Furthermore, the elements b2 a p−n−3m a n with m = 0, 1, . . . , (N − 3)/3, and n = 0, 1, . . . , N − 1, belong to the same conjugacy class. Since cba p−n−3m a n c−1 = b2 a −n a −p+n+3m , these elements belong to the same conjugacy class for Δ(6N 2 ).

(13.41)

132

13

Δ(6N 2 )

Similarly to what was done for Δ(6N 2 ) when N/3 = integer, we can study the conjugacy classes of elements involving c. It turns out that these conjugacy classes are the same for the group Δ(6N 2 ) when N/3 = integer and when N/3 is an integer. Here we summarize the conjugacy classes of Δ(6N 2 ) when N/3 is an integer: C1 :

{e},  p −p  , p = N/3, 2N/3, a a  k −k −2k −k k 2k  (k) C3 : , k = 1, . . . , N − 1, k = N/3, 2N/3, a a ,a a ,a a   m m− − −m −m −m − m− m  −m  (,m) , a a ,a a ,a a ,a a ,a a ,a a C6 : , m = 0, . . . , N − 1,   (q) C2N 2 /3 : ba q−n−3m a n , b2 a −n a n+3m−q  n = 0, 1, . . . , N − 1,  N −3 , m = 0, 1, . . . , 3 q = 0, 1, 2,    () ca +n a n , b2 ca − a −−n , bca −n a   n = 0, 1, . . . , N − 1 , C3N :  = 0, . . . , N − 1, (13.42) (,m) do not satisfy (13.6). The order h of each element in the given where , m in C6 conjugacy classes is found to be (p) C1 :

C1 :

h = 1,

(p)

C1 :

h = 3,

(k) C3 :

h = N/ gcd(N, k),

(,m)

C6

:

(q)

h = 3,

()

h = 2N/ gcd(N, ).

C2N 2 /3 : C3N :

(13.43)

h = N/ gcd(N, , m),

(p)

(q)

(k) (,m) () , C2N 2 /3 , and C3N The numbers of each of the conjugacy classes C1 , C3 , C6

are 2, N − 3, (N 2 − 3N + 6)/6, 3, and N , respectively. The total number of conjugacy classes is thus equal to (N 2 + 9N + 24)/6. The relations (2.18) and (2.19) for Δ(6N 2 ) with N/3 integer are m1 + 22 m2 + 32 m3 + · · · = 6N 2 , m1 + m2 + m3 + · · · =

N2

+ 9N + 24 . 6

The solution is (m1 , m2 , m3 , m6 ) = (2, 4, 2(N − 1), [N (N − 3)]/6).

(13.44) (13.45)

13.2

Δ(6N 2 ) with N/3 Integer

133

13.2.2 Characters and Representations There are two singlets 1k . Their characters are the same as those of the group Δ(6N 2 ) with N/3 integer. That is, because c2 = e, the characters of the two singlets have two possible values χ1k (c) = (−1)k with k = 0, 1, and they correspond to two singlets 1k . Note that χ1k (a) = χ1k (a  ) = χ1k (b) = 1. The group Δ(6N 2 ) with N/3 integer has four doublets 2k with k = 1, 2, 3, 4. One of them, namely 21 , is the same as the doublet (13.19) of Δ(6N 2 ) with N/3 = integer. The other three doublet representations will be obtained following the discussion of the sextet representations. We thus consider the sextet representations, which are the same as those of Δ(6N 2 ) with N/3 integer, viz., (13.20) and (13.21), i.e.,  b=

b1 0

 0 , b2

 c=

 0 1 , 1 0

 a=

a1 0

 0 , a2

a =

  a1 0

 0 , a2 (13.46)

where ⎛ ⎞ 0 1 0 b1 = ⎝0 0 1⎠ , 1 0 0 ⎛ l ⎞ ρ 0 0 −1 a1 = a  2 = ⎝ 0 ρ k 0 ⎠, −l−k 0 0 ρ

⎛

⎞ 0 0 1 b2 = ⎝ 1 0 0 ⎠ , 0 1 0 ⎛ l+k ρ 0 −1 a2 = a  1 = ⎝ 0 ρ −l 0 0

(13.47) ⎞ 0 0 ⎠.

ρ −k

(13.48) In certain cases, the above representation becomes reducible. For N/3 an integer, 6[[k],[]] is reducible if (k, ) = (N/3, N/3), (2N/3, 2N/3), k +  = 0(mod N ), k = 0,  = 0, and  = 0, k = 0. Thus, the number of irreducible sextet representations is [N 2 − 2 − N − 2(N − 1)]/6. Among the above reducible sextet representations, the irreducible triplets are obtained in the same way as (13.25) for Δ(6N 2 ) with N/3 = integer, i.e., ⎛

0 1 b(31k ) = ⎝0 0 1 0 ⎛ k ρ 0 a(31k ) = ⎝ 0 ρ −k 0 0

⎞ 0 1⎠ , 0 ⎞ 0 0⎠ , 1

⎛ 0 c(31k ) = ⎝0 1 ⎛ 1  ⎝ a (31k ) = 0 0

⎞ 0 1 1 0⎠ , 0 0 0 ρk 0

b(32k ) = b(31k ),

c(32k ) = −c(31k ),

a(32k ) = a(31k ),

a  (32k ) = a  (31k ).

⎞ 0 0 ⎠,

(13.49)

ρ −k

(13.50)

134

13

Δ(6N 2 )

In addition, one can obtain the three doublets by diagonalizing the above reducible sextet representations:

b(22 ) = b(21 ),

b(23 ) = b(21 ),

b(24 ) = 1,



c(22 ) = c(21 ),

ω2 a(22 ) = a (22 ) = 0

c(23 ) = c(21 ),

ω a(23 ) = a (23 ) = 0







a(24 ) = a  (24 ) =

c(24 ) = c(21 ),



ω 0

 0 , ω

 0 , ω2

 0 . ω2

(13.51)

(13.52)

(13.53)

The characters are shown in Table 13.2.

13.2.3 Tensor Products

When 3N is an integer, the number of doublets is increased. Since products with other representations are the same as those for 3N = integer, it suffices to examine products with additional doublets: ⎞ x,−k− ⎜ xk, ⎟ ⎟ ⎜ ⎜x−k−,k ⎟ ⎟ ⎜ ⎜xk+,− ⎟ ⎟ ⎜ ⎝ x−,−k ⎠ x−k,k+ 6 ⎛

⎞ x,−k− y1 ⎜ ωxk, y1 ⎟ ⎟ ⎜ 2   ⎜ω x−k−,k y1 ⎟ y ⎟ ⊗ 1 =⎜ ⎜ xk+,− y2 ⎟ y2 2 ⎟ ⎜ 2 ⎝ ωx−,−k y2 ⎠ ω2 x−k,k+ y2 6 [[k],[]] [[k+2N/3],[+2N/3]] ⎞ ⎛ x,−k− y2 ⎜ ω2 xk, y2 ⎟ ⎟ ⎜ ⎜ωx−k−,k y2 ⎟ ⎟ ⎜ ⊕⎜ , ⎟ ⎜ xk+,− y1 ⎟ ⎝ω2 x−,−k y1 ⎠ ωx−k,k+ y1 6 ⎛

[[k+N/3],[+N/3]]

(13.54)

(m)

C1

1

3

2N/ gcd(N, m)

(m) C3N

(−1)r

1

N/ gcd(N, m, n) 1

N/ gcd(N, m)

1

1

3

1

(τ ) C2N 2 /3

C6

(m,n)

C3

(m)

C1

χ 1r

h

0

−1

2

2

2

2

χ 21

0

ω2+τ + ω(4+2τ )

ωm+n + ω2m+2n

2

2

2

χ 22

Table 13.2 Characters of Δ(6N 2 ) for N/3 integer

0

ω1+τ + ω2+2τ

ωm+n + ω2m+2n

2

2

2

χ 23

0

ωτ + ω2τ

ωm+n + ω2m+2n

2

2

2

χ2 4

−ρ −mk

ρ mk + ρ −nk + ρ (−m+n)k

ρ mk + ρ −nk + ρ (−m+n)k

0

ρ −2mk + 2ρ mk

ρ −2mk + 2ρ mk

ρ −mk

ρ −2mk + 2ρ mk

0

3

3

χ32k

ρ −2mk + 2ρ mk

χ31k

0

0

ρ mk+n + ρ (−m+n)k−m + ρ −nk+(m−n) + ρ −nk−m + ρ mk+(m−n) + ρ (−m+n)k+n

2ρ m(k−) + 2ρ −m(2k+) + 2ρ m(k+2)

2ρ m(k−) + 2ρ −m(2k+) + 2ρ m(k+2)

6

χ6[[k],[]]

13.2 Δ(6N 2 ) with N/3 Integer 135

136

13

⎛

⎞ x,−k− ⎜ xk, ⎟ ⎜ ⎟ ⎜x−k−,k ⎟ ⎜ ⎟ ⎜xk+,− ⎟ ⎜ ⎟ ⎝ x−,−k ⎠ x−k,k+ 6

⎞ x,−k− y1 ⎜ ωxk, y1 ⎟ ⎟ ⎜ 2   ⎜ω x−k−,k y1 ⎟ y ⎟ ⊗ 1 =⎜ ⎜ xk+,− y2 ⎟ y2 2 ⎟ ⎜ 3 ⎝ ωx−,−k y2 ⎠ ω2 x−k,k+ y2 6 [[k],[]] [[k+N/3],[+N/3]] ⎞ ⎛ x,−k− y2 ⎜ ω2 xk, y2 ⎟ ⎟ ⎜ ⎜ωx−k−,k y2 ⎟ ⎟ ⎜ ⊕⎜ , ⎟ ⎜ xk+,− y1 ⎟ 2 ⎝ω x−,−k y1 ⎠ ωx−k,k+ y1 6 [[k+2N/3],[+2N/3]] ⎛ ⎞ ⎞ ⎛ x,−k− x,−k− y1 ⎜ xk, ⎟ ⎜ xk, y1 ⎟ ⎜ ⎟ ⎟ ⎜   ⎜x−k−,k ⎟ ⎜x−k−,k y1 ⎟ y1 ⎜ ⎟ ⎟ ⎜ ⊗ = ⎜xk+,− ⎟ ⎜xk+,− y2 ⎟ y2 2 ⎜ ⎟ ⎟ ⎜ 4 ⎝ x−,−k ⎠ ⎝ x−,−k y2 ⎠ x−k,k+ 6 x−k,k+ y2 6 [[k],[]] [[k+N/3],[+N/3]] ⎞ ⎛ x,−k− y2 ⎜ xk, y2 ⎟ ⎟ ⎜ ⎜x−k−,k y2 ⎟ ⎟ ⎜ ⊕⎜ , ⎟ ⎜xk+,− y1 ⎟ ⎝ x−,−k y1 ⎠ x−k,k+ y1 6

Δ(6N 2 )

⎛

(13.55)

(13.56)

[[k+2N/3],[+2N/3]]

⎞ xk,0 y1 ⎜ ωx−k,k y1 ⎟ ⎟ ⎜ 2   ⎜ω x0,−k y1 ⎟ y1 ⎟ ⎜ ⊗ =⎜ ⎟ y2 2 ⎜ x0,−k y2 ⎟ 2 ⎝ ωx−k,k y2 ⎠ ω2 xk,0 y2 6

,

(13.57)

⎞ xk,0 y1 ⎜ ωx−k,k y1 ⎟ ⎟ ⎜ 2   ⎜ ω x0,−k y1 ⎟ y1 ⎟ ⎜ ⊗ =⎜ y2 2 −x0,−k y2 ⎟ ⎟ ⎜ 2 ⎝−ωx−k,k y2 ⎠ −ω2 xk,0 y2 6

,

(13.58)

⎛

⎛

⎞ xk,0 ⎝x−k,k ⎠ x0,−k 3

1k

[[−k+2N/3],[k+2N/3]]

⎛

⎛

⎞ xk,0 ⎝x−k,k ⎠ x0,−k 3

2k

[[−k+2N/3],[k+2N/3]]

13.2

Δ(6N 2 ) with N/3 Integer

137

⎞ xk,0 y1 ⎜ ωx−k,k y1 ⎟ ⎟ ⎜ 2   ⎜ω x0,−k y1 ⎟ y ⎟ ⊗ 1 =⎜ ⎜ x0,−k y2 ⎟ y2 2 ⎟ ⎜ 3 ⎝ ωx−k,k y2 ⎠ ω2 xk,0 y2 6 ⎛

⎛

⎞ xk,0 ⎝x−k,k ⎠ x0,−k 3

1k

,

(13.59)

,

(13.60)

[[−k+N/3],[k+N/3]]

⎛

⎞ xk,0 y1 ⎜ ωx−k,k y1 ⎟ ⎜ 2 ⎟   ⎜ ω x0,−k y1 ⎟ y ⎟ ⊗ 1 =⎜ ⎜ −x0,−k y2 ⎟ y2 2 ⎜ ⎟ 3 ⎝−ωx−k,k y2 ⎠ −ω2 xk,0 y2 6

⎞ xk,0 ⎝x−k,k ⎠ x0,−k 3 ⎛

2k

[[−k+N/3],[k+N/3]]

⎞ xk,0 y1 ⎜x−k,k y1 ⎟ ⎟ ⎜   ⎜x0,−k y1 ⎟ y1 ⎟ ⎜ ⊗ =⎜ y2 2 x0,−k y2 ⎟ ⎟ ⎜ 4 ⎝x−k,k y2 ⎠ xk,0 y2 6 ⎛

⎛

⎞ xk,0 ⎝x−k,k ⎠ x0,−k 3

1k

(13.61)

,

[[−k+N/3],[k+N/3]]

⎞ xk,0 y1 ⎜ x−k,k y1 ⎟ ⎟ ⎜   ⎜ x0,−k y1 ⎟ y1 ⎟ ⎜ ⊗ =⎜ y2 2 −x0,−k y2 ⎟ ⎟ ⎜ 4 ⎝−x−k,k y2 ⎠ −xk,0 y2 6 ⎛

⎛

⎞ xk,0 ⎝x−k,k ⎠ x0,−k 3

2k

(13.62)

.

[[−k+N/3],[k+N/3]]

The tensor products among doublets are as follows:           y x y x1 ⊗ 1 = x1 y2 + x2 y1 1 ⊕ x1 y2 − x2 y1 1 ⊕ 2 2 , 0 1 x2 2 y2 2 x1 y1 2 k

for k = 1, 2, 3, 4,

k

(13.63)

k

        x1 y1 x2 y2 x1 y2 ⊗ = ⊕ , x2 2 y2 2 x1 y1 2 x2 y1 2 1 2 3 4         x1 y x y x y ⊗ 1 = 2 2 ⊕ 2 1 , x2 2 y2 2 x1 y1 2 x1 y2 2 1 3 2 4         y x y x y x1 ⊗ 1 = 1 2 ⊕ 1 1 , x2 2 y2 2 x2 y1 2 x2 y2 2 1

4

2

3

(13.64) (13.65) (13.66)

138

13

        x1 y x y x y ⊗ 1 = 2 2 ⊕ 1 2 , x2 2 y2 2 x1 y1 2 x2 y1 2 2 3 1 4         y x y x y x1 ⊗ 1 = 1 1 ⊕ 1 2 , x2 2 y2 2 x2 y2 2 x2 y1 2 2 4 1 3         x1 y x y x y ⊗ 1 = 1 2 ⊕ 1 1 . x2 2 y2 2 x2 y1 2 x2 y2 2 3

4

1

Δ(6N 2 )

(13.67) (13.68) (13.69)

2

Tensor products of non-trivial singlets with additional doublets are the same as those of 2 for N/3 = integer.

13.3 Δ(54) Here we consider a simple example of Δ(6N 2 ). The group Δ(6) is nothing but S3 and Δ(24) is isomorphic to S4 . Thus, the simplest non-trivial example is Δ(54).

13.3.1 Conjugacy Classes All elements of Δ(54) can be written in the form bk c a m a n , where k, m, n = 0, 1, 2, and  = 0, 1. Half of them are the elements of Δ(27), whose conjugacy classes are (1) shown in (10.37). Since cac−1 = a −1 and ca  c−1 = a −1 , the conjugacy classes C1 (2) and C1 of Δ(27) still correspond to the conjugacy classes of Δ(54). However, the conjugacy classes C3(0,1) and C3(0,2) of Δ(27) are combined into one class. Similarly, (2,p  )

since cba k a  c−1 = b2 ca k a  c−1 , the conjugacy classes C3 and C3 of Δ(27) for p + p  = 0 (mod 3) are combined into one class of Δ(54). Let us consider the conjugacy classes of elements involving c. For example, we obtain (1,p)

  a k a  ca m a −k a − = ca m+p a p ,

(13.70)

where p = −k − . Thus, the element ca m is conjugate to ca m+p a p with p = 0, 1, 2. Furthermore, it is found that   b ca m+p a p b−1 = b2 ca −m a −m−p ,   b ca −m a −m−p b−1 = bca −p a m . These elements thus belong to the same conjugacy class.

(13.71) (13.72)

13.3

139

Δ(54)

Using the above results, the elements of Δ(54) are classified into the following conjugacy classes: C1 : (1)

C1 : (2)

C1 : (0,1)

C6

(0) C6 :

C6(1) : (2)

C6 : (0) C9 : (1)

C9 : (2)

C9 :

:

{e}, h = 1,  2  h = 3, aa ,  2  h = 3, a a ,    h = 3, a , a, a 2 a 2 , a 2 , a 2 , aa  ,   h = 3, b, ba 2 a  , baa 2 , b2 , b2 a 2 a  , b2 aa 2 ,   h = 3, ba, ba  , ba 2 a 2 , b2 a 2 , b2 a 2 , b2 aa  ,   2 h = 3, ba , baa  , ba 2 , b2 a, b2 a 2 a 2 , b2 a  ,   p p 2 −p  ca a , b ca , bca −p  p = 0, 1, 2 , h = 2,    1+p p 2 2 −1−p ca a , b ca a , bca −p a   p = 0, 1, 2 , h = 6,    2+p p 2 ca a , b caa −2−p , bca −p a 2  p = 0, 1, 2 , h = 6.

(13.73)

The total number of conjugacy classes is equal to ten. The relations (2.18) and (2.19) for Δ(54) lead to m1 + 22 m2 + 32 m3 + · · · = 54,

(13.74)

m1 + m2 + m3 + · · · = 10.

(13.75)

The solution is (m1 , m2 , m3 ) = (2, 4, 4), whence there are two singlets, four doublets, and four triplets.

13.3.2 Characters and Representations We start by discussing the two singlets. It is straightforward to show that χ1k (a) = χ1k (a  ) = χ1k (b) = 1 for the two singlets from the above conjugacy class structure. In addition, since c2 = e, the two values are χ1k (c) = (−1)k with k = 0, 1. These correspond to the two singlets. Next, we consider the triplets. For example, the generators a, a  , b, and c can be represented by ⎛ ⎛ k ⎞ ⎞ 1 0 0 ω 0 0 0 ⎠, a = ⎝ 0 ω2k 0⎠ , a  = ⎝0 ωk 0 0 ω2k 0 0 1 (13.76) ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 1 b = ⎝0 0 1⎠ , c = ⎝0 1 0⎠ , 1 0 0 1 0 0

140

13

Δ(6N 2 )

Table 13.3 Characters of Δ(54) χ10

χ 11

χ 21

χ2 2

χ2 3

χ2 4

χ311

χ312

χ321

χ322

C1

1

1

2

2

2

2

3

3

3

3

C1(1) (2) C1 C6(0,1) (0) C6 C6(1) C6(2) (0) C9 C9(1) (2) C9

1

1

2

2

2

2

3ω

3ω2

3ω

3ω2

1

1

2

2

2

2

3ω2

3ω

3ω2

3ω

1

1

2

−1

−1

−1

0

0

0

0

1

1

−1

−1

−1

2

0

0

0

0

1

1

−1

2

−1

−1

0

0

0

0

1

1

−1

−1

2

−1

0

0

0

0

1

−1

0

0

0

0

1

1

−1

−1

1

−1

0

0

0

0

ω2

ω

−ω2

−ω

1

−1

0

0

0

0

ω

ω2

−ω

−ω2

on 31k for k = 1, 2. Obviously, the Δ(54) algebra is still satisfied when c is replaced by −c. That is, the generators a, a  , b, and c are represented by ⎞ ⎛ k 0 0 ω a = ⎝ 0 ω2k 0⎠ , 0 0 1 ⎛ ⎞ 0 1 0 b = ⎝0 0 1⎠ , 1 0 0

⎛ 1 a  = ⎝0 0 ⎛ 0 c=⎝ 0 −1

⎞ 0 0 ⎠, ω2k ⎞ 0 −1 −1 0 ⎠ , 0 0 0 ωk 0

(13.77)

on 32k for k = 1, 2. The characters χ3 for 31k and 32k are shown in Table 13.3. Now, consider the doublets. There are four doublet representations of Δ(54). The generators a, a  , b, and c are represented by 

 1 0 , a=a = 0 1   2 0 ω a = a = , 0 ω   ω 0 , a = a = 0 ω2   ω 0 , a = a = 0 ω2 



ω b= 0  ω b= 0  ω b= 0  1 b= 0

 0 , ω2  0 , ω2  0 , ω2  0 , 1



0 c= 1  0 c= 1  0 c= 1  0 c= 1

The characters χ2 for 21,2,3,4 are shown in Table 13.3.

 1 , on 21 , 0  1 , on 22 , 0  1 , on 23 , 0  1 , on 24 . 0

(13.78) (13.79) (13.80) (13.81)

13.3

141

Δ(54)

13.3.3 Tensor Products The tensor products between triplets are as follows: ⎛ ⎞ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ y1 x1 y1 x2 y3 + x3 y2 x2 y 3 − x 3 y 2 x1 ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = ⎝x2 y2 ⎠ ⊕ ⎝x3 y1 + x1 y3 ⎠ ⊕ ⎝x3 y1 − x1 y3 ⎠ , x3 3 y3 3 x3 y3 3 x1 y2 + x2 y1 3 x1 y2 − x2 y1 3 11

11

12

12

22

(13.82) ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎛ ⎞ y1 x1 y1 x2 y3 + x3 y2 x2 y 3 − x 3 y 2 x1 ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = ⎝x2 y2 ⎠ ⊕ ⎝x3 y1 + x1 y3 ⎠ ⊕ ⎝x3 y1 − x1 y3 ⎠ , x3 3 y3 3 x3 y3 3 x1 y2 + x2 y1 3 x1 y2 − x2 y1 3 12

12

11

11

21

(13.83) ⎞ ⎞ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎛ x1 y1 x1 y1 x2 y3 + x3 y2 x2 y 3 − x 3 y 2 ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = ⎝x2 y2 ⎠ ⊕ ⎝x3 y1 + x1 y3 ⎠ ⊕ ⎝x3 y1 − x1 y3 ⎠ , x3 3 y3 3 x3 y3 3 x1 y2 + x2 y1 3 x1 y2 − x2 y1 3 21

21

12

12

22

(13.84) ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎛ ⎞ y1 x1 y1 x2 y3 + x3 y2 x2 y 3 − x 3 y 2 x1 ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = ⎝x2 y2 ⎠ ⊕ ⎝x3 y1 + x1 y3 ⎠ ⊕ ⎝x3 y1 − x1 y3 ⎠ , x3 3 y3 3 x3 y3 3 x1 y2 + x2 y1 3 x1 y2 − x2 y1 3 22

22

11

11

21

(13.85) ⎛ ⎞ ⎛ ⎞   y1 x1 2   ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = x1 y1 + x2 y2 + x3 y3 ⊕ x1 y1 + ω x22 y2 + ωx3 y3 10 ωx1 y1 + ω x2 y2 + x3 y3 2 1 x3 3 y3 3 11 12   x y + ω2 x2 y3 + ωx3 y1 ⊕ 1 2 ωx1 y3 + ω2 x2 y1 + x3 y2 2 2   2 x y + ω x2 y1 + ωx3 y2 ⊕ 1 3 ωx1 y2 + ω2 x2 y3 + x3 y1 2 3   x y + x2 y1 + x3 y2 ⊕ 1 3 , (13.86) x1 y2 + x2 y3 + x3 y1 2 4

⎛ ⎞ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ y1 x1 y1 x3 y 2 − x 2 y 3 x3 y2 + x2 y3 x1 ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = ⎝x2 y2 ⎠ ⊕ ⎝x1 y3 − x3 y1 ⎠ ⊕ ⎝x1 y3 + x3 y1 ⎠ , x3 3 y3 3 x3 y3 3 x2 y1 − x1 y2 3 x2 y1 + x1 y2 3 11 21 22 12 22 (13.87)

142

13

Δ(6N 2 )

⎛ ⎞ ⎛ ⎞   y1 x1 2   ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = x1 y1 + x2 y2 + x3 y3 ⊕ x1 y1 + ω x22y2 + ωx3 y3 11 −ωx1 y1 − ω x2 y2 − x3 y3 2 1 x3 3 y3 3 11 22   x1 y2 + ω2 x2 y3 + ωx3 y1 ⊕ −ωx1 y3 − ω2 x2 y1 − x3 y2 2 2   2 x1 y3 + ω x2 y1 + ωx3 y2 ⊕ −ωx1 y2 − ω2 x2 y3 − x3 y1 2 3   x1 y3 + x2 y1 + x3 y2 ⊕ , (13.88) −x1 y2 − x2 y3 − x3 y1 2 4

⎛ ⎞ ⎛ ⎞   x1 y1 2   ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = x1 y1 + x2 y2 + x3 y3 ⊕ x1 y1 + ω x22y2 + ωx3 y3 11 −ωx1 y1 − ω x2 y2 − x3 y3 2 1 x3 3 y3 3 12 21   x1 y3 + ω2 x2 y1 + ωx3 y2 ⊕ −ωx1 y2 − ω2 x2 y3 − x3 y1 2 2   2 x1 y2 + ω x2 y3 + ωx3 y1 ⊕ −ωx1 y3 − ω2 x2 y1 − x3 y2 2 3   x1 y2 + x2 y3 + x3 y1 ⊕ . (13.89) −x1 y3 − x2 y1 − x3 y2 2 4

⎛ ⎞ ⎛ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎞ y1 x1 y1 x3 y2 − x2 y3 x3 y 2 + x 2 y 3 x1 ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = ⎝x2 y2 ⎠ ⊕ ⎝x1 y3 − x3 y1 ⎠ ⊕ ⎝x1 y3 + x3 y1 ⎠ , x3 3 y3 3 x3 y3 3 x2 y1 − x1 y2 3 x2 y1 + x1 y2 3 12

22

21

11

21

(13.90) ⎛ ⎞ ⎛ ⎞   y1 x1 2   ⎝x2 ⎠ ⊗ ⎝y2 ⎠ = x1 y1 + x2 y2 + x3 y3 ⊕ x1 y1 + ω x22 y2 + ωx3 y3 10 ωx1 y1 + ω x2 y2 + x3 y3 2 1 x3 3 y3 3 21 22   x y + ω2 x2 y3 + ωx3 y1 ⊕ 1 2 ωx1 y3 + ω2 x2 y1 + x3 y2 2 2   2 x y + ω x1 y3 + ωx2 y1 ⊕ 3 2 x1 y2 + ωx2 y3 + ω2 x3 y1 2 3   x y + x2 y1 + x3 y2 ⊕ 1 3 , (13.91) x1 y2 + x2 y3 + x3 y1 2 4

13.3

143

Δ(54)

The tensor products between doublets are:           y x y x1 ⊗ 1 = x1 y2 + x2 y1 1 ⊕ x1 y2 − x2 y1 1 ⊕ 2 2 , 0 1 x2 2 y2 2 x1 y1 2 k

k

(13.92)

k

for k = 1, 2, 3, 4,         x1 y1 x2 y2 x1 y2 ⊗ = ⊕ , x2 2 y2 2 x1 y1 2 x2 y1 2

(13.93)

        y1 x2 y2 x2 y1 x1 ⊗ = ⊕ , x2 2 y2 2 x1 y1 2 x1 y2 2

(13.94)

        x1 y x y x y ⊗ 1 = 1 2 ⊕ 1 1 , x2 2 y2 2 x2 y1 2 x2 y2 2

(13.95)

        y1 x2 y2 x1 y2 x1 ⊗ = ⊕ , x2 2 y2 2 x1 y1 2 x2 y1 2

(13.96)

        x1 y1 x1 y1 x1 y2 ⊗ = ⊕ , x2 2 y2 2 x2 y2 2 x2 y1 2

(13.97)

        y x y x y x1 ⊗ 1 = 1 2 ⊕ 1 1 . x2 2 y2 2 x2 y1 2 x2 y2 2

(13.98)

1

1

1

2

2

3

2

3

3

2

4

2

3

1

4

1

4

1

4

4

3

4

3

2

The tensor products between doublets and triplets are: ⎛ ⎞ ⎛ ⎛ ⎞ ⎞   y1 x 1 y1 + ω 2 x2 y1 x1 y1 − ω2 x2 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ωx1 y2 + ωx2 y2 ⎠ ⊕ ⎝ωx1 y2 − ωx2 y2 ⎠ , (13.99) x2 2 1 y3 3 ω2 x1 y3 + x2 y3 3 ω2 x1 y3 − x2 y3 3 1k

1k

2k

⎛ ⎞ ⎛ ⎛ ⎞ ⎞   y1 x1 y1 + ω2 x2 y1 x1 y1 − ω2 x2 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ωx1 y2 + ωx2 y2 ⎠ ⊕ ⎝ωx1 y2 − ωx2 y2 ⎠ , (13.100) x2 2 1 y3 3 ω 2 x1 y3 + x2 y3 3 ω 2 x 1 y 3 − x2 y 3 3 2k

2k

1k

⎛ ⎞ ⎛ ⎛ ⎞ ⎞   ωx1 y2 + x2 y3 ωx1 y2 − x2 y3 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ω2 x1 y3 + ω2 x2 y1 ⎠ ⊕ ⎝ω2 x1 y3 − ω2 x2 y1 ⎠ , x2 2 2 y3 3 x1 y1 + ωx2 y2 x1 y1 − ωx2 y2 311 321 11 (13.101)

144

13

Δ(6N 2 )

⎛ ⎞ ⎛ ⎛ ⎞ ⎞   ωx1 y2 + x2 y3 ωx1 y2 − x2 y3 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ω2 x1 y3 + ω2 x2 y1 ⎠ ⊕ ⎝ω2 x1 y3 − ω2 x2 y1 ⎠ , x2 2 2 y3 3 x1 y1 + ωx2 y2 x1 y1 − ωx2 y2 321 311 21 (13.102) ⎛ ⎞ ⎛ ⎛ ⎞ ⎞   ωx1 y3 + x2 y2 ωx1 y3 − x2 y2 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ω2 x1 y1 + ω2 x2 y3 ⎠ ⊕ ⎝ω2 x1 y1 − ω2 x2 y3 ⎠ , x2 2 2 y3 3 x1 y2 + ωx2 y1 x1 y2 − ωx2 y1 312 322 12 (13.103) ⎛ ⎞ ⎛ ⎛ ⎞ ⎞   ωx1 y3 + x2 y2 ωx1 y3 − x2 y2 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ω2 x1 y1 + ω2 x2 y3 ⎠ ⊕ ⎝ω2 x1 y1 − ω2 x2 y3 ⎠ , x2 2 2 y3 3 x1 y2 + ωx2 y1 x1 y2 − ωx2 y1 322 312 22 (13.104) ⎛ ⎞ ⎛ ⎛ ⎞ ⎞   ωx1 y3 + x2 y2 ωx1 y3 − x2 y2 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ω2 x1 y1 + ω2 x2 y3 ⎠ ⊕ ⎝ω2 x1 y1 − ω2 x2 y3 ⎠ , x2 2 3 y3 3 x1 y2 + ωx2 y1 x1 y2 − ωx2 y1 311 321 11 (13.105) ⎞ ⎞ ⎛ ⎞ ⎛ ⎛   ωx1 y3 + x2 y2 ωx1 y3 − x2 y2 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ω2 x1 y1 + ω2 x2 y3 ⎠ ⊕ ⎝ω2 x1 y1 − ω2 x2 y3 ⎠ , x2 2 3 y3 3 x1 y2 + ωx2 y1 x1 y2 − ωx2 y1 321 311 21 (13.106) ⎛ ⎞ ⎛ ⎛ ⎞ ⎞   ωx1 y2 + x2 y3 ωx1 y2 − x2 y3 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ω2 x1 y3 + ω2 x2 y1 ⎠ ⊕ ⎝ω2 x1 y3 − ω2 x2 y1 ⎠ , x2 2 3 y3 3 x1 y1 + ωx2 y2 x1 y1 − ωx2 y2 312 322 12 (13.107) ⎛ ⎞ ⎛ ⎛ ⎞ ⎞   ωx1 y2 + x2 y3 ωx1 y2 − x2 y3 y1 x1 ⊗ ⎝y2 ⎠ = ⎝ω2 x1 y3 + ω2 x2 y1 ⎠ ⊕ ⎝ω2 x1 y3 − ω2 x2 y1 ⎠ , x2 2 3 y3 3 x1 y1 + ωx2 y2 x1 y1 − ωx2 y2 322 312 22 (13.108) ⎛ ⎞ ⎛ ⎛ ⎞ ⎞   y1 x1 y3 + x2 y2 x1 y3 − x2 y2 x1 ⊗ ⎝y2 ⎠ = ⎝x1 y1 + x2 y3 ⎠ ⊕ ⎝x1 y1 − x2 y3 ⎠ , (13.109) x2 2 4 y3 3 x1 y2 + x2 y1 3 x1 y2 − x2 y1 3 11 11 21 ⎛ ⎞ ⎛ ⎛ ⎞ ⎞   y1 x1 y3 + x2 y2 x1 y3 − x2 y2 x1 ⊗ ⎝y2 ⎠ = ⎝x1 y1 + x2 y3 ⎠ ⊕ ⎝x1 y1 − x2 y3 ⎠ , (13.110) x2 2 4 y3 3 x1 y2 + x2 y1 3 x1 y2 − x2 y1 3 21 21 11 ⎛ ⎞ ⎛ ⎛ ⎞ ⎞   y1 x1 y2 + x2 y3 x1 y2 − x2 y3 x1 ⊗ ⎝y2 ⎠ = ⎝x1 y3 + x2 y1 ⎠ ⊕ ⎝x1 y3 − x2 y1 ⎠ , (13.111) x2 2 4 y3 3 x1 y1 + x2 y2 3 x1 y1 − x2 y2 3 12

12

22

References

145

⎛ ⎞ ⎛ ⎛ ⎞ ⎞   y1 x1 y2 + x2 y3 x1 y 2 − x 2 y 3 x1 ⊗ ⎝y2 ⎠ = ⎝x1 y3 + x2 y1 ⎠ ⊕ ⎝x1 y3 − x2 y1 ⎠ . x2 2 4 y3 3 x1 y1 + x2 y2 3 x1 y1 − x2 y2 3 22

22

(13.112)

12

Finally, the tensor products of the non-trivial singlet 11 with other representations are: 2 k ⊗ 1 1 = 2k ,

31k ⊗ 11 = 32k ,

32k ⊗ 11 = 31k .

(13.113)

References 1. Escobar, J.A., Luhn, C.: J. Math. Phys. 50, 013524 (2009). arXiv:0809.0639 [hep-th]

Chapter 14

Subgroups and Decompositions of Multiplets

In particle physics, a symmetry is often broken to a subgroup to describe low energy phenomena. Therefore, it is very important to study the breaking patterns of discrete groups and decompositions of multiplets. In this chapter, we discuss decompositions of multiplets for the groups studied in the previous chapters. Suppose that a finite group G has order N and that M is a divisor of N . Then, Lagrange’s theorem implies that a finite group H with order M is a candidate for a subgroup of G (see Appendix A). An irreducible representation r G of G can  be decomposed into irreducible representations r H,m of its subgroup H as r G = m r H,m . If the trivial singlet of H is included in such a decomposition m r H,m , and a scalar field with such a trivial singlet develops its vacuum expectation value (VEV), the group G breaks to H . On the other hand, if a scalar field in a multiplet r G develops its VEV and it does not correspond to the trivial singlet of H , the group G breaks not to H , but to another group. In the following sections, we consider decompositions of multiplets of G into multiplets of subgroups. For a finite group G, there are several chains of subgroups, viz., G → G1 → · · · → Gk → ZN → {e}, G → G1 → · · · → Gm → ZM → {e}, and so on. It should be obvious that the smallest non-trivial subgroup in these chains will be an Abelian group such as ZN or ZM . Since we concentrate on subgroups discussed explicitly in the previous chapters, we consider the largest subgroup, i.e., G1 or G1 , in each chain of subgroups.

14.1 S3 We begin with S3 because it is the smallest non-Abelian discrete group, with order equal to 2 × 3 = 6. There are then two candidates for subgroups. One is a group of H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_14, © Springer-Verlag Berlin Heidelberg 2012

147

148

14

Subgroups and Decompositions of Multiplets

order two and the other a group of order three. The former corresponds to Z2 and the latter to Z3 . As discussed in Sect. 3.1, S3 consists of {e, a, b, ab, ba, bab}, where a 2 = e and (ab)3 = e. Now the subgroup Z2 consists of, e.g., {e, a}, or another combination such as {e, b} or {e, bab}, which also correspond to Z2 . The subgroup Z3 consists of {e, ab, ba = (ab)2 }. The group S3 has two singlets 1 and 1 , and one doublet 2. Both subgroups Z2 and Z3 are Abelian. Thus, decompositions of multiplets under Z2 and Z3 are rather simple. We shall examine these decompositions in what follows. The breaking pattern of S3 is summarized in Table 14.1.

14.1.1 S3 → Z3 The elements {e, ab, ba} of S3 constitute the Z3 subgroup, which is a normal subgroup. There is no other choice to obtain a Z3 subgroup. There are three singlet representations 1k with k = 0, 1, 2 for Z3 , that is, ab = ωk on 1k . Recall that χ1 (ab) = χ1 (ab) = 1 for both 1 and 1 of S3 . Thus, both 1 and 1 of S3 correspond to 10 of Z3 . On the other hand, the doublet 2 of S3 decomposes into two singlets of Z3 . Since χ2 (ab) = −1, the S3 doublet 2 decomposes into 11 and 12 of Z3 . In order to understand this decomposition explicitly, we take the two-dimensional representation of the group element ab as given in (2.28), viz., √   −1/2 − 3/2 ab = √ . (14.1) 3/2 −1/2 Then the doublet (x1 , x2 ) decomposes into two non-trivial singlets, viz., 11 : x1 − ix2 ,

12 : x1 + ix2 .

(14.2)

14.1.2 S3 → Z2 The subgroup Z2 of S3 consists of, e.g., {e, a}. It has two singlet representations 1k , for k = 0, 1, that is, a = (−1)k on 1k . Recall that χ1 (a) = 1 and χ1 (a) = −1 for 1 and 1 of S3 . Thus, 1 and 1 of S3 correspond to 10 and 11 of Z2 , respectively. On the other hand, the doublet 2 of S3 decomposes into two singlets of Z2 . Since χ2 (a) = −1, the S3 doublet 2 decomposes into 10 and 11 of Z2 . Indeed, the element a is represented on 2 in (2.28) by   1 0 a= . (14.3) 0 −1 Then for the doublet (x1 , x2 ), the elements x1 and x2 correspond to x1 = 10 and x2 = 11 , respectively.

14.2

149

S4

Table 14.1 Breaking pattern of S3

S3

Z3

S3

Z2

1

10

1

10

1

10

1

11

2

11 + 12

2

10 + 11

In addition to {e, a}, there are other Z2 subgroups, viz., {e, b} and {e, aba}. In both cases, the same results are obtained when we choose a proper basis. These are examples of Abelian subgroups. For non-Abelian subgroups, the same situation arises. That is, different elements of a finite group G can generate the same subgroup. We exemplify this by considering D6 . All elements of D6 can be written in the form a m bk for m = 0, 1, . . . , 5, and k = 0, 1, where a 6 = e and bab = a −1 . Denoting a˜ = a 2 , we find that the elements a˜ m bk for m = 0, 1, 2, and k = 0, 1, correspond to the subgroup D3  S3 . On the other hand, denoting b˜ = ab, we find that the elements a˜ m b˜ k for m = 0, 1, 2, and k = 0, 1, correspond to another D3 subgroup. The decompositions of D6 multiplets into D3 multiplets are the same for both D3 subgroups when we move to a proper basis.

14.2 S4 As mentioned in Chap. 13, the group S4 is isomorphic to Δ(24) and (Z2 × Z2 )  S3 . It is convenient to use the terminology of (Z2 ×Z2 )S3 . That is, all the elements are expressed in the form bk c a m a n with k = 0, 1, 2, and , m, n = 0, 1 (see Chap. 13). The generators a, a  , b, and c are related to the notation of Sect. 3.2 in the following way: b = c1 ,

c = f1 ,

a = a4 ,

a  = a2 .

(14.4)

They satisfy the algebraic relations b3 = c2 = (bc)2 = a 2 = a 2 = e, bab−1 = a −1 a −1 , cac−1 = a −1 ,

ba  b−1 = a,

aa  = a  a, (14.5)

ca  a −1 = a −1 .

Furthermore, their representations on 1, 1 , 2, 3, and 3 are shown in Table 14.2. As subgroups, S4 contains non-Abelian groups S3 , A4 , and Σ(8), the latter being (Z2 × Z2 )  Z2 . Thus, the decompositions of S4 are non-trivial compared with those of S3 . The breaking pattern of S4 is summarized in Table 14.3.

150

14

Subgroups and Decompositions of Multiplets

Table 14.2 Representations of S4 elements 1

1

2 

b

1

1 

c

1

−1

a

1

1

a

1

1

0

0

ω2

0 1

1 0

1 0



⎛



⎞

⎟ 0 1⎠ 0 0 ⎞ 0 1 ⎟ 1 0⎠ 1 0 0

⎛



S3

S4

1

1

1

1

2

0 1 0

⎞

⎜ ⎟ ⎝0 0 1⎠ 1 0 0 ⎛ ⎞ 0 0 −1 ⎜ ⎟ ⎝ 0 −1 0 ⎠ ⎞

−1 0 0 ⎜ ⎟ ⎝ 0 −1 0 ⎠ 0 0 1 ⎛ ⎞ 1 0 0 ⎜ ⎟ ⎝ 0 −1 0 ⎠ 0 0 −1

0 1

S4

0 1 0

⎜ ⎝0 1 ⎛ 0 ⎜ ⎝0

0 1 

Table 14.3 Breaking pattern of S4

ω

⎛



1 0 

3

3

−1

0 ⎞ −1 0 0 ⎜ ⎟ ⎝ 0 −1 0 ⎠ 0 0 1 ⎛ ⎞ 1 0 0 ⎜ ⎟ ⎝ 0 −1 0 ⎠ 0 0 −1 ⎛

0

A4

S4

1

1

1

1+0

1

1

1

1−0

2

2

1 + 1

2

1+0 + 1−0

3

1+2

3

3

3

1+1 + 2

3

1

3

3

3

1−1 + 2

Table 14.4 Representations of S3 elements

+2

1

1

Σ(8)

2 

b

1

1 

c

1

−1

ω

0

0 0

ω2  1

1

0



14.2.1 S4 → S3 The subgroup S3 elements are {a1 , b1 , d1 , d1 , e1 , f1 }. Alternatively, they can be denoted by bk c with k = 0, 1, 2, and  = 0, 1, i.e., {e, b, b2 , c, bc, b2 c}. Among them, Table 14.4 shows the representations of the generators b and c on 1, 1 , and 2 of S3 . Then the singlets 1 and 1 and the doublet 2 remain the same representation of S3 , i.e., 1, 1 , and 2 for each. Triplets 3 and 3 are decomposed to 1 + 2 and 1 + 2. The

14.2

151

S4

Table 14.5 Representations of A4 elements

1

1

1

3 ⎛

b

1

ω

ω2

a

1

1

1

1

1

1

⎞

⎜ ⎟ ⎝0 0 1⎠ 1 0 0 ⎛ ⎞ −1 0 0 ⎜ ⎟ ⎝ 0 −1 0 ⎠ ⎛

a

0 1 0

0 1

0 0

⎜ ⎝ 0 −1 0 0

1 0

⎞

⎟ 0 ⎠ −1

components of 3 (x1 , x2 , x3 ) decompose to 1 and 2 according to 

2 : x1 + ω2 x2 + ωx3 , ωx1 + x2 + ω2 x3 , 1 : x1 + x2 + x3 ,

(14.6)

and the components of 3 decompose to 1 and 2 according to 

1 : x1 + x2 + x3 , 2 : x1 + ω2 x2 + ωx3 , −ωx1 − x2 − ω2 x3 .

(14.7)

14.2.2 S4 → A4 The A4 subgroup consists of bk a m a n with k = 0, 1, 2, and m, n = 0, 1. Recall that A4 is isomorphic to Δ(12). Table 14.5 shows the representations of the generators b, a, and a  on 1, 1 , 1 , and 3 of A4 . Then the representations 1, 1 , 2, 3, and 3 of S4 decompose to 1, 1, 1 + 1 , 3, and 3, respectively.

14.2.3 S4 → Σ(8) The subgroup Σ(8), i.e., (Z2 × Z2 )  Z2 , consists of elements c a m a n with , m, n = 0, 1. Table 14.6 shows the representations of the generators c, a, and a  on 1+0 , 1+1 , 1−0 , 1−1 , and 21,0 of Σ(8). The representations 1, 1 , 2, 3, and 3 of S4 decompose to 1+0 , 1−0 , 1+0 + 1−0 , 1+1 + 2, and 1−1 + 2, respectively. The components of 3 (x1 , x2 , x3 ) decompose to 1+1 and 2 according to 1+1 : x2 , and the components of

3

2 : (x3 , x1 ),

(14.8)

decompose to 1−1 and 2 according to 1−1 : x2 ,

2 : (x3 , −x1 ).

(14.9)

152

14

Table 14.6 Representations of Σ(8) elements

1+0

Subgroups and Decompositions of Multiplets 1+1

1−0

1−1

21,0 

c

1

−1

1

−1 

a

1

−1

1

−1 

a

1

−1

Table 14.7 Representations of a, ˜ a˜  , and b˜ in Δ(12)

1

1k

−1

1

a

1

Table 14.8 Breaking pattern of A4

1

0



1

0

0

−1

−1

0

0

1

 

−1 0

0

⎞

⎜ ⎟ ⎝ 0 1 0 ⎠ 0 0 −1 ⎛ ⎞ −1 0 0 ⎜ ⎟ ⎝ 0 −1 0 ⎠ ⎛

0

0

0 1 0

⎞

1

⎜ ⎟ ⎝0 0 1⎠ 1 0 0

ωk

b

1

3 ⎛

a

0

Z3

A4

Z2 × Z2

1k

1k

1k

10,0

3

10 + 11 + 12

3

11,1 + 10,1 + 11,0

A4

14.3 A4 The group A4 is isomorphic to Δ(12). Here, we apply the generic results of Δ(3N 3 ) to the group A4 . All elements of Δ(12) can be written in the form bk a m a n with k = 0, 1, 2, and m, n = 0, 1. Table 14.7 shows the representations of generators a, a  , and b on each representation. Regarding subgroups, A4 contains Abelian groups Z3 and Z2 × Z2 . The breaking pattern of A4 is summarized in Table 14.8.

14.3.1 A4 → Z3 The group Z3 consists of {e, b, b2 }. The representations 1k and 3 of Δ(12) decompose to 1k and 10 + 11 + 12 , respectively. Decomposition of the triplet (x1 , x2 , x3 ) is obtained by 10 : x1 + x2 + x3 , 11 : x1 + ω2 x2 + ωx3 , and 12 : x1 + ωx2 + ω2 x3 .

14.4

153

A5

Table 14.9 Breaking pattern of A5

A5

A4

A5

D5

A5

D3

1

1

1

1+

1

1+

3

3

3

1− + 21

3

1− + 2

3

3

3

1− + 22

3

1− + 2

4

1+3

4

21 + 22

4

1+ + 1− + 2

5

1 + 1 + 3

5

1+ + 21 + 22

5

1+ + 2 + 2

14.3.2 A4 → Z2 × Z2 The subgroup Z2 × Z2 consists of {e, a, a  , aa  }. The representations 1k and 3 of A4 decompose to 10,0 and 11,1 + 10,1 + 11,0 , respectively.

14.4 A5 All elements of A5 can be expressed as products of s = a and t = bab, as shown in Sect. 4.2. Regarding subgroups, A5 contains the non-Abelian groups A4 , D5 , and S3 . The breaking pattern of A5 is summarized in Table 14.9.

14.4.1 A5 → A4 The subgroup A4 has elements {e, b, a, ˜ bab ˜ 2 , b2 ab, ˜ ba, ˜ ab, ˜ ab ˜ a, ˜ b2 a, ˜ b2 ab ˜ ab}, ˜ 2 ˜ where a˜ = ab aba. We denote t = b and s˜ = a. ˜ These satisfy the relations s˜ 2 = t˜ 3 = (˜s t˜)3 = e,

(14.10)

and correspond to the generators s and t of the group A4 in Sect. 4.1. The representations 1, 3, 3 , 4, and 5 of A5 decompose to 1, 3, 3, 1 + 3, and 1 + 1 + 3, respectively.

14.4.2 A5 → D5 The subgroup D5 consists of the elements a k a˜ m with k = 0, 1, and m = 0, 1, 2, 3, 4, where a˜ ≡ bab2 a. These satisfy a 2 = a˜ 5 = e and a aa ˜ = a˜ 4 . In order to identify the 2 ˜ D5 basis used in Chap. 6, we define b = abab a. Table 14.10 shows the representations of these generators a˜ and b˜ on 1+ , 1− , 21 , and 22 of D5 . The representations 1, 3, 3 , 4, and 5 of A5 then decompose to 1+ , 1− + 21 , 1− + 22 , 21 + 22 , and 1+ + 21 + 22 , respectively.

154

14

Subgroups and Decompositions of Multiplets

Table 14.10 Representations of elements of D5 1+

1−

21 

a˜

1

1 

b˜

−1

1

22

exp 2πi/5



0



− exp 2πi/5

0  0 1



1 0

exp 4πi/5



0 − exp 4πi/5

0  0 1 1 0

Table 14.11 Representations of T  1 1 1 2

2 

s 1 1 1

− √i

3

 r 1 1 1  t 1 ω ω2

√  2

1 √ 2

−1

0

0

−1

ω

0

0

ω2

−1 



2 

− √i

3





√  2

1 √ 2

−1

0

0

−1

ω2

0

0

1



−1 

3 

− √i

3





−1

0

0

−1

1

0

0

ω

√  2

1 √ 2



−1 

⎛ 1 3

⎛

−1

⎜ ⎝ 2 2

1 ⎜ ⎝0 0 ⎛ 1 ⎜ ⎝0 0

0 1 0 0 ω 0

2 −1 2 ⎞ 0 ⎟ 0⎠ 1 ⎞ 0 ⎟ 0 ⎠ ω2

2

⎞

⎟ 2 ⎠ −1

14.4.3 A5 → S3  D3 Recall that the group S3 is isomorphic to D3 . The subgroup D3 consists of elements bk a˜ m with k = 0, 1, 2, and m = 0, 1, where we define a˜ = ab2 ab2 ab. These gener˜ a˜ = b2 . The representations 1, 3, 3 , 4, and 5 of A5 then ators satisfy a˜ 2 = e and ab decompose to 1+ , 1− + 2, 1− + 2, 1+ + 1− + 2, and 1+ + 2 + 2, respectively.

14.5 T  All elements of T  can be expressed in terms of the generators s, t, and r, which satisfy the algebraic relations s 2 = r, r 2 = t 3 = (st)3 = e, and rt = tr. Table 14.11 shows the different representations of s, t, and r. Regarding subgroups, T  contains Z6 , Z4 , and Q4 . The breaking pattern of T  is summarized in Table 14.12.

14.5.1 T  → Z6 The subgroup Z6 consists of elements a m , with m = 0, . . . , 5, where a = rt and a 6 = e. The group Z6 has six singlet representations 1n with n = 0, . . . , 5. On the

14.6

General DN

Table 14.12 Breaking pattern of T 

155 T  Z6

T  Z4

T

1

10

1

10

1

1++

1

12

1

10

1

1++

1 10

1

1++

1 14

Q4

2

11 + 15

2

11 + 13

2

2

2

13 + 15

2

11 + 13

2

2

2 11 + 13

2

2

3

1+− + 1−+ + 1−−

2 13 + 15 3

10 + 12 + 14

3

10 + 12 + 12

singlet 1n , the generator a is represented by a = e2πin/6 . The representations 1, 1 , 1 , 2, 2 , 2 , and 3 of T  thus decompose to 10 , 12 , 14 , 15 + 11 , 15 + 13 , 13 + 15 , and 10 + 12 + 14 , respectively.

14.5.2 T  → Z4 The subgroup Z4 consists of elements {e, s, s 2 , s 3 }. Z4 has four singlet representations 1m with m = 0, 1, 2, 3. On the singlet 1m , the generator s is represented by s = eπim/2 . All the doublets 2, 2 , and 2 of T  decompose to two singlets 11 and 13 of Z4 according to √ √ 1+ 3 −1 + 3 11 : √ x1 + x2 , ix1 + x2 , 13 : − √ 2 2 where (x1 , x2 ) correspond to the doublets. In addition, the triplet 3 : (x1 , x2 , x3 ) decomposes to singlets 10 + 12 + 12 according to 10 : (x1 + x2 + x3 ), 12 : (−x1 + x3 ), and 12 : (−x1 + x2 ).

14.5.3 T  → Q4 We consider the subgroup Q4 , which consists of elements s m bk with m = 0, 1, 2, 3, and k = 0, 1. The generator b is defined by b = tst 2 . The representations 1, 1 , 1 , 2, 2 , 2 , and 3 of T  decompose to 1++ , 1++ , 1++ , 2, 2, 2, and 1+− + 1−+ + 1−− , respectively.

14.6 General DN Since the group DN is isomorphic to ZN  Z2 , DM and ZN appear as subgroups of DN in addition to Z2 , where M is a divisor of N . Recall that all elements of DN can

156

14

Subgroups and Decompositions of Multiplets

Table 14.13 Breaking pattern of DN for even N DN

Z2

DN

ZN

1++

10

1++

10

1+−

10

1+−

1N/2

1−+

11

1−+

1N/2

1−−

11

1−−

10

2k

10 + 11

2k

1k + 1N −k

DN

DM (M is even)

DN

DM (M is odd)

1++

1++

1++

1+

1+−

1++ (M/N is even)

1+−

1+

1+− (M/N is odd)

1−+

1−

1−− (M/N is even)

1−−

1−

1−+ (M/N is odd)

2k

2k (k = k  + Mn) 2˜ M−k  (k = M − k  )

1−+ 1−−

1−−

2k

2k (k = k  + Mn) 2˜ M/2−k (k = M/2 − k  + Mn)

1+ + 1− (k = Mn)

1+− + 1−+ (k = M(2n + 1)/2) 1++ + 1−− (k = Mn)

be written in the form a m bk with m = 0, . . . , N − 1, and k = 0, 1. There are singlets and doublets 2k , where k = 1, . . . , N/2 − 1, for N even and k = 1, . . . , (N − 1)/2, for N odd. On the doublet 2k , the generators a and b are represented by     k 0 0 1 ρ , b= , (14.11) a= 1 0 0 ρ −k where ρ = e2πi/N . For N even, there are four singlets 1±± . The generator b is represented by b = 1 on 1+± , while b = −1 on 1−± . The generator a is represented by a = 1 on 1++ and 1−− , while a = −1 on 1+− and 1+− . For N odd, there are two singlets 1± . The generator b is represented by b = 1 on 1+ and b = −1 on 1− , while a = 1 on both singlets. The general breaking patterns of DN are summarized in Table 14.13 for even N and Table 14.14 for odd N .

14.6.1 DN → Z2 The two elements e and b generate the Z2 subgroup. Obviously, there are two singlet representations 10 and 11 , where the subscript denotes the Z2 charge. That is, we have b = 1 on 10 and b = −1 on 11 .

14.6

General DN

Table 14.14 Breaking pattern of DN for odd N

157 DN

Z2

DN ZN

DN

DM

1+

10

1+

10

1+

1+

1−

10

1−

10

1−

1−

2k

10 + 11

2k

1k + 1N −k

2k

2k (N = k  + Mn) 2˜ M−k  (k = Mn − k  ) 1+ + 1− (k = Mn)

When N is even, the singlets 1++ and 1+− of DN become 10 of Z2 and the singlets 1−+ and 1−− of DN become 11 of Z2 . The doublets 2k of DN , viz., (x1 , x2 ), decompose to two singlets according to 10 : x1 + x2 and 11 : x1 − x2 . When N is odd, the singlet 1+ of DN becomes 10 of Z2 and the singlet 1− of DN becomes 11 of Z2 . The decompositions of doublets 2k are the same as for N even.

14.6.2 DN → ZN The subgroup ZN consists of the elements {e, a, . . . , a N −1 }. Obviously, it is a normal subgroup of DN and there are N types of irreducible singlet representation 10 , 11 , . . . , 1N −1 . On 1k , the generator a is represented by a = ρ k . When N is even, the singlets 1++ and 1−− of DN become 10 of ZN and the singlets 1+− and 1−+ of DN become 1N/2 of ZN . The doublets 2k and (x1 , x2 ) decompose to two singlets according to 1k : x1 and 1N −k : x2 . When N is odd, both 1+ and 1− of DN become 10 of ZN . The decompositions of doublets 2k are the same as for N even.

14.6.3 DN → DM The above decompositions of DN are rather straightforward, because the subgroups are Abelian. Here we consider the subgroup DM , where M is a divisor of N . The decompositions of DN to DM are expected to be non-trivial. We denote a˜ = a  , where  = N/M and  is therefore an integer. The subgroup DM consists of elements a˜ m bk with m = 0, . . . , M − 1, and k = 0, 1. There are three relevant combinations of (N, M), i.e., (N, M) = (even, even), (even, odd), and (odd, odd). We start with the combination (N, M) = (even, even). Recall that ab of DN is represented by ab = 1 on 1±+ and ab = −1 on 1±− . Thus, the representations of a  b depend on whether  is even or odd. When  is odd, ab and a  b are represented in the same way on each of the above singlets. On the other hand, when  is even, we always have the singlet representations with a  = 1. The doublets 2k of DN correspond to the doublets 2k  of DM when k = k  (mod M). In addition, when k =

158

14

Subgroups and Decompositions of Multiplets

−k  (mod M), the doublets 2k (x1 , x2 ) of DN correspond to the doublets 2M−k  (x2 , x1 ) of DM . That is, the components are swapped over and we denote it by 2˜ M−k  . Furthermore, the other doublets 2k of DN decompose to two singlets of DM according to 1+− + 1−+ with 1+− : x1 + x2 and 1−+ : x1 − x2 for k = (M/2) (mod M) and 1++ + 1−− with 1++ : x1 + x2 and 1−− : x1 − x2 for k = 0 (mod M). Next we consider the case (N, M) = (even, odd). In this case, the singlets 1++ , 1+− , 1−+ , and 1−− of DN become 1+ , 1+ , 1− , and 1− of DM , respectively. The doublets 2k of DN correspond to the doublets 2k  of DM when k = k  (mod M). In addition, when k = −k  (mod M), the doublets 2k (x1 , x2 ) of DN correspond to the doublets 2M−k  (x2 , x1 ) of DM . Furthermore, when k = 0 (mod M), the other doublets 2k of DN decompose to two singlets of DM according to 1+ + 1− , where 1+ : x1 + x2 and 1− : x1 − x2 . We now consider the case (N, M) = (odd, odd). In this case, the singlets 1+ and 1− of DN become 1+ and 1− of DM . The doublets 2k of DN correspond to the doublets 2k  of DM when k = k  (mod M). In addition, when k = −k  (mod M), the doublets 2k (x1 , x2 ) of DN correspond to the doublets 2M−k  (x2 , x1 ) of DM . Furthermore, when k = 0 (mod M), the other doublets 2k of DN decompose to two singlets of DM according to 1+ + 1− , where 1+ : x1 + x2 and 1− : x1 − x2 .

14.7 D4 Here we study D4 , which is the second smallest discrete symmetry. All elements of D4 can be written in the form a m bk with m = 0, 1, 2, 3, and k = 0, 1. Since the order of D4 is 8, it contains order 2 and 4 subgroups. There are two types of order 4 group, corresponding to Z2 × Z2 and Z4 . All subgroups are Abelian so the decompositions are rather simple.

14.7.1 D4 → Z4 The subgroup Z4 consists of the elements {e, a, a 2 , a 3 }. Obviously, it is a normal subgroup of D4 and there are four types of irreducible singlet representation 1m with m = 0, 1, 2, 3, where a is represented by a = eπim/2 . From the characters of the group D4 , it is found that singlet representations 1++ and 1−− of D4 correspond to 10 of Z4 , while 1+− and 1−+ of D4 correspond to 12 of Z4 . For the D4 doublet 2, it is convenient to use the diagonal basis for the matrix a, so that   i 0 a= . (14.12) 0 −i Then we can read off that the doublet 2 : (x1 , x2 ) decomposes to two singlets according to 11 : x1 and 13 : x2 .

14.8

General QN

159

Table 14.15 Representations of QN for N = 4n (N = 4n)

1++

1+−

1−+

1−−

2k=odd 

a

1

−1

−1

1 

b

1

1

−1

−1

2k=even

ρk

0

0

ρ −k 

0

i

i

0



 

ρk

0

0

ρ −k 

0 1



1 0

14.7.2 D4 → Z2 × Z2 We denote a˜ = a 2 . Then the subgroup Z2 × Z2 consists of elements {e, a, ˜ b, ab}, ˜ where ab ˜ = ba˜ and a˜ 2 = b2 = e. Their representations are clearly quite simple, that is, 1±± , whose Z2 × Z2 charges are determined by a˜ = ±1 and b = ±1. We use the notation that the first (second) subscript of 1±± denotes the Z2 charge for a˜ (b). Then the singlets 1++ and 1+− of D4 correspond to 1++ of Z2 × Z2 , while 1−+ and 1−− of D4 correspond to 1+− of Z2 × Z2 . The doublet 2 of D4 decomposes to 1−+ and 1−− of Z2 . In addition to the above, there is another choice of Z2 × Z2 subgroup which consists of the elements {e, a 2 , ab, a 3 b}. In this case, we obtain the same decomposition of D4 .

14.7.3 D4 → Z2 Furthermore, both Z4 and Z2 × Z2 include the subgroup Z2 . The decomposition of D4 to Z2 is rather straightforward.

14.8 General QN Recall that all elements of QN can be written in the form a m bk with m = 0, . . . , N − 1, and k = 0, 1, where a N = e and b2 = a N/2 . Similarly to DN with N even, there are four singlets 1±± and doublets 2k with k = 1, . . . , N/2 − 1. Tables 14.15 and 14.16 show the representations of a and b on these representations for N = 4n and N = 4n + 2. In general, the group QN includes Z4 , ZN , and QM as subgroups. The breaking patterns of QN are summarized in Table 14.17 for N = 4n and Table 14.18 for N = 4n + 2.

160

14

Subgroups and Decompositions of Multiplets

Table 14.16 Representations of QN for N = 4n + 2 (N = 4n + 2)

1++

1+−

1−+

1−−

2k=odd 

−1

1

a

−1

1 

1

b

−i

i

−1

2k=even

ρk

0

0

ρ −k 

0

i

i

0



 

ρk

0

0

ρ −k 

0 1



1 0

Table 14.17 Breaking pattern of QN for N = 4n. All parameters n, m, k, k  , n are integers QN

Z4

QN

ZN

1++

10

1++

10

1+−

10

1+−

1N/2

1−+

12

1−+

12

1−−

12

1−−

10

2k

11 + 13

2k

1k + 1N −k

QN

QM (M = 4m)

QN

QM (M = 4m + 2)

1++

1++

1++

1++

1+−

1++ (M/N is even)

1+−

1++

1+− (M/N is odd)

1−+

1−−

1−− (M/N is even)

1−−

1−−

1−+ (M/N is odd)

2k

2k (k = k  + Mn ) 2˜ M−k  (k = Mn − k  )

1−+ 1−−

1−−

2k

2k (k = k  + Mn ) 2˜ M−k (k = Mn − k  ) 1+− + 1−+ (k

= M(2n

1+− + 1−+ (k = M(2n + 1)/2) 1++ + 1−− (k = Mn ) + 1)/2)

1++ + 1−− (k = Mn )

14.8.1 QN → Z4 First we consider the subgroup Z4 , which consists of the elements {e, b, b2 , b3 }. Obviously, there are four singlet representations 1m for Z4 , and the generator b is represented by b = eπim/2 on 1m . When N = 4n, 1++ and 1+− of QN correspond to 10 of Z4 , while 1−+ and 1−− of QN correspond to 12 of Z4 . The doublets 2k of QN , viz., (x1 , x2 ), decompose to two singlets 11 : (x1 − ix2 ) and 13 : (x1 + ix2 ). When N = 4n + 2, 1++ , 1+− , 1−+ , and 1−− of QN correspond to 10 , 11 , 12 , and 13 of Z4 , respectively. The decompositions of doublets 2k are the same as for N = 4n.

14.8

General QN

161

Table 14.18 Breaking pattern of QN for N = 4n + 2 and M = 4m + 2. All parameters n, m, k, k  , n are integers QN

Z4

QN

ZN

QN

QM

1++

10

1++

10

1++

1++

1+−

10

1+−

1N/2

1+−

1++

1−+

12

1−+

1N/2

1−+

1−−

1−−

12

1−−

10

1−−

1−−

2k

11 + 13

2k

1k + 1N −k

2k

2k (N = k  + Mn ) 2˜ M−k  (N = Mn − k  ) 1+− + 1−+ (N = M(2n + 1)/2) 1++ + 1−− (N = Mn )

14.8.2 QN → ZN We now consider the subgroup ZN , which consists of the elements {e, a, . . . , a N −1 }. Obviously, it is normal subgroup of QN and there are N types of irreducible singlet representation 10 , 11 , . . . , 1N −1 . On the singlet 1m of ZN , the generator a is represented by a = ρ m . The singlets 1++ and 1−− of QN correspond to 10 of ZN and the singlets 1+− and 1−+ of QN correspond to 1N/2 of ZN . The doublets 2k and (x1 , x2 ) of QN decompose to two singlets 1k : x1 and 1N −k : x2 .

14.8.3 QN → QM We consider the subgroup QM , where M is a divisor of N . We define a˜ = a  with  = N/M, where  is thus an integer. The subgroup QM consists of all elements a˜ m bk with m = 0, . . . , M − 1, and k = 0, 1. There are three relevant combinations (N, M), i.e., (N, M) = (4n, 4m), (4n, 4m + 2), and (4n + 2, 4m + 2). We start with the combination (N, M) = (4n, 4m), where  = N/M can be even or odd. Recall that ab of QN is represented by ab = 1 on 1±+ and ab = −1 on 1±− . Thus, the representations of a  b depend on whether  is even or odd. When  is odd, ab and a  b are represented in the same way as on each of the above singlets. On the other hand, when  is even, we always have the singlet representations with a  = 1. The doublets 2k of QN correspond to the doublets 2k  of QM when k = k  (mod M). In addition, when k = −k  (mod M), the doublets 2k (x1 , x2 ) of QN correspond to the doublets 2M−k  (x2 , x1 ) of QM . Furthermore, the other doublets 2k of QN decompose to two singlets of QM according to 1+− + 1−+ with 1+− : x1 + x2 and 1−+ : x1 − x2 for k = (M/2) (mod M) and 1++ + 1−− with 1++ : x1 + x2 and 1−− : x1 − x2 for k = 0 (mod M). Next we consider the case (N, M) = (4n, 4m + 2), where  must be even. Similarly to the above case with  even, the singlets 1++ , 1+− , 1−+ , and 1−− of QN correspond to 1++ , 1++ , 1−− , and 1−− of QM . The results for decompositions

162

14

Subgroups and Decompositions of Multiplets

of doublets are also the same as for the above case with (N, M) = (4n, 4m) and  = N/M even. Next, we consider the case (N, M) = (4n + 2, 4m + 2), where  must be odd. In this case, the results for decompositions are the same as for the case with (N, M) = (4n, 4m) and  = N/M odd.

14.9 Q4 All elements of Q4 can be expressed in the form a m bk with m = 0, 1, 2, 3, and k = 0, 1. Since the order of Q4 is equal to 8, it contains order 2 and 4 subgroups. There are several order 4 subgroups which correspond to Z4 groups.

14.9.1 Q4 → Z4 For example, the elements {e, a, a 2 , a 3 } comprise one Z4 subgroup. It is clearly a normal subgroup of Q4 and there are four types of irreducible singlet representation 1m with m = 0, 1, 2, 3, where a is represented by a = eπim/2 . From the characters of the group Q4 , it is found that 1++ and 1−− of Q4 correspond to 10 of Z4 , while 1−+ and 1+− of Q4 correspond to 12 of Z4 . For the doublets of Q4 , it is convenient to use the diagonal basis for the matrix a so that   i 0 a= . (14.13) 0 −i Then we find that the doublet 2 (x1 , x2 ) decomposes to two singlets 11 : x1 and 13 : x2 . Other Z4 subgroups can be found, namely, {e, b, b2 , b3 } and {e, ab, (ab)2 , (ab)3 }. For these Z4 subgroups, we obtain the same results when we choose a proper basis. Furthermore, Z2 subgroups can be found from the above Z4 groups. The decomposition of Z4 to Z2 is rather straightforward.

14.10 QD2N Since the group QD2N is isomorphic to ZN  Z2 , DM and ZN appear as subgroups of DN in addition to Z2 , where M is a divisor of N . Recall that all elements of QD2N can be expressed in the form a m bk with m = 0, . . . , N − 1, and k = 0, 1. There are four singlets and (N/2 − 1) doublets 2k , where k = 1, . . . , N/2 − 1. On the doublet 2k , the generators a and b are represented by  k    ρ 0 0 1 a= , b= , (14.14) 1 0 0 ρ k(N/2−1)

14.10

QD2N

163

Table 14.19 Breaking pattern of QD2N QD2N

Z2

QD2N

ZN

QD2N

DN/2

1++

10

1++

10

1++

1++

1−+

10

1−+

1N/2

1−+

1+−

1+−

11

1+−

10

1+−

1−−

1−−

11

1−−

1N/2

1−−

1−+

2k

10 + 11

2k

1k + 1k(N/2−1)

2k

2k (k = k  + N/4) 1+− + 1−+ (k = N/4)

where ρ = e2πi/N . There are four singlets 1ss  with s, s  = ±. The generator a is represented by a = 1 on 1+s  , while a = −1 on 1−s  . The generator b is represented by b = 1 on 1s+ , while b = −1 on 1s− . The general breaking pattern of QD2N is summarized in Table 14.19.

14.10.1 QD2N → Z2 The two elements e and b generate the Z2 subgroup. Obviously, there are two singlet representations 10 and 11 , where the subscript denotes the Z2 charge. That is, we have b = 1 on 10 and b = −1 on 11 . The singlets 1++ and 1−+ of QD2N become 10 of Z2 , while the singlets 1+− and 1−− of QD2N become 11 of Z2 . The doublets 2k (x1 , x2 ) of QD2N decompose to two singlets 10 : x1 + x2 and 11 : x1 − x2 .

14.10.2 QD2N → ZN The subgroup ZN consists of the elements {e, a, . . . , a N −1 }. Obviously it is a normal subgroup of DN and there are N types of irreducible singlet representation 10 , 11 , . . . , 1N −1 . On the 1k , the generator a is represented by a = ρ k . The singlets 1++ and 1+− of QD2N become 10 of ZN , while the singlets 1−+ and 1−− of QD2N become 1N/2 of ZN . The doublets 2k and (x1 , x2 ) decompose to two singlets 1k : x1 and 1k(N/2−1) : x2 .

14.10.3 QD2N → DN/2 The above decompositions of QD2N are rather straightforward because the subgroups are Abelian. The decomposition from QD2N to DN/2 is expected to be nontrivial. We define a˜ = a 2 . The subgroup DN/2 consists of all elements of the form a˜ m bk with m = 0, . . . , N/2 − 1, and k = 0, 1.

164

14

Table 14.20 Representations of Σ(2N 2 )

Subgroups and Decompositions of Multiplets

1+n

1−n

ρn

ρn

2p,q 

a

 a

ρn

ρn 

b

1

−1

ρq

0

0

ρp

ρp

0

0

ρq 

0

1

1

0

 

The singlet representations 1st decompose to 1tu of DN/2 with u = st. The doublets 2k of QD2N correspond to the doublets 2k  of DN/2 when k = k  (mod N/2). In addition, the doublet 2N/4 of DN decomposes to two singlets of DN/2 according to 1+− + 1−+ with 1+− : x1 + x2 and 1−+ : x1 − x2 .

14.11 General Σ(2N 2 ) Recall that all elements of the group Σ(2N 2 ) can be written in the form bk a m a n with k = 0, 1, and m, n = 0, 1, . . . , N − 1. The generators a, a  , and b satisfy a N = a N = b2 = e, aa  = a  a, and bab = a  , that is, a, a  , and b correspond to  , and Z of (Z × Z  )  Z , respectively. Table 14.20 shows the differZN , ZN 2 N 2 N ent representations of these generators. The number of doublets 2p,q is equal to N(N − 1)/2 with the relation p > q. In general, the group Σ(2N 2 ) contains Z2N , ZN × ZN , DN , QN , and Σ(2M 2 ) as subgroups. The breaking pattern of Σ(2N 2 ) is summarized in Table 14.21.

14.11.1 Σ(2N 2 ) → Z2N The group Σ(2N 2 ) always includes a subgroup Z2N . We consider the elements of Z2N as (ba)m with m = 0, . . . , 2N − 1. There are 2N singlet representations 1m for Z2N and the generator ba is represented by b = ρ m on 1m , where ρ = eπi/N . The representations 1+n , 1−n , and 2,m of Σ(2N 2 ) then decompose to 12n , 12n+N , and 1+m + 1+m+N , respectively. The components of doublets (x , xm ) correspond to 1+m : (ρ  x + ρ m xm ) and 1+m+N : (ρ  x − ρ m xm ).

14.11.2 Σ(2N 2 ) → ZN × ZN The subgroup ZN × ZN consists of the elements a m a n with m, n = 0, . . . , N − 1. Obviously, it is a normal subgroup of Σ(2N 2 ). There are N 2 singlet representations

14.11

General Σ(2N 2 )

165

Table 14.21 Breaking pattern of Σ(2N 2 ). All parameters n, m, k, k  , n are integers Σ(2N 2 )

Z2N

Σ(2N 2 )

ZN × ZN

1+n

12n

1+n

1n,n

1−n

12n+N

1−n

1n,n

2,m

1+m + 1+m+N

2,m

1,m + 1m,

Σ(2N 2 )

DN (N is even)

Σ(2N 2 )

DN (N is odd)

1+n

1++

1+n

1+

1−n

1−−

1−n

1−

2,m

2 2˜ k ( = m − k  )

2,m

2k ( = m + k  ) 2˜ N −k ( = m − k  )

k

( = m + k  )

1+− + 1−+ ( = m + N/2) Σ(2N 2 ) 1+n

1−n 2,m

QN

Σ(2N 2 )

Σ(2M 2 )

1++ (n is even)

1+n

1+n

1−− (n is odd)

1−n

1−n

1−− (n is even)

2,m

2 ,m ( =  + Mn, m = m + Mn )

1++ (n is odd) 2k ( = m + k  ) 1+− + 1−+ ( = m + N/2)

1m,n and the generators a and a  are represented by a = ρ m and a  = ρ n on 1m,n . The representations 1+n , 1−n , and 2,m of Σ(2N 2 ) then decompose to 1n,n , 1n,n , and 1,m + 1m, , respectively.

14.11.3 Σ(2N 2 ) → DN We now consider DN as a subgroup of Σ(2N 2 ). We define a˜ = a −1 a  . Then the subgroup DN consists of the elements a˜ m bk with k = 0, 1, and m = 0, . . . , N − 1. Table 14.22 shows the representations of the generators a˜ and b on each representation of Σ(2N 2 ). First we consider the case where N is even. The doublets 2p,q of Σ(2N 2 ) are still doublets of DN , except when p − q = N/2. On the other hand, when p − q = N/2, the doublets decompose to two singlets of DN . The representations 1+n , 1−n , 2q+k  ,q , 2q−k  ,q , and 2q+N/2,q of Σ(2N 2 ) then decompose to 1++ , 1−− , 2k  , 2˜ k  , and 1+− + 1−+ , respectively.

166

14

Table 14.22 Representations of a˜ and b in Σ(2N 2 )

1+n

Subgroups and Decompositions of Multiplets 1−n

2p,q 

a˜

1

1 

b

Table 14.23 Representations of a˜ and b˜ in Σ(2N 2 )

1

−1

1+n

1−n

1

1 

b˜

1

−1

0 0

1

1

0

0 



ρ −(p−q)

2p,q 

a˜

ρ p−q



ρ p−q

0

0 0

ρ −(p−q)  (−1)q

(−1)p

0

Next, we consider the case where N is odd. In this case, the representations 1+n , 1−n , 2q+k  ,q , and 2q−k  ,q of Σ(2N 2 ) decompose to 1+ , 1− , 2k , and 2˜ N −k  , respectively.

14.11.4 Σ(2N 2 ) → QN We consider QN as a subgroup of Σ(2N 2 ) with N even. We define a˜ = a −1 a  and b˜ = ba N/2 . Then the subgroup QN consists of all elements of the form a˜ m b˜ k with m = 0, . . . , N − 1, and k = 0, 1. Table 14.23 shows the various representations of these generators a˜ and b˜ for each representation of Σ(2N 2 ). The singlets 1+n and 1−n of Σ(2N 2 ) then become singlets of QN , namely, 1++ and 1−− for even n and 1−− and 1++ for odd n. The decompositions of doublets are obtained in a similar way to the decomposition for Σ(2N 2 ) → DN , whence 2q+k  ,q and 2q+N/2 go to 2k and 1+− + 1−+ , respectively.

14.11.5 Σ(2N 2 ) → Σ(2M 2 ) We consider the subgroup Σ(2M 2 ), where M is a divisor of N . We denote a˜ = a  and a˜  = a  with  = N/M, whence  is an integer. The subgroup Σ(2M 2 ) consists of all elements of the form bk a˜ m a˜ n with k = 0, 1, and m, n = 0, . . . , M − 1. Table 14.24 shows the representations of a, ˜ a˜  , and b˜ on each representation 2 of Σ(2N ). The representations 1+n , 1−n , and 2p+Mn,q+Mn of Σ(2N 2 ) then correspond to the representations 1+n , 1−n , and 2p,q of Σ(2M 2 ), where n, n are integers.

14.12

167

Σ(32)

Table 14.24 Representations of a, ˜ a˜  , and b˜ in Σ(2N 2 )

1+n

1−n

ρ n

ρ n

2p,q 

a˜

 a˜ 

ρ n

ρ n  −1

1

b



ρ q

0

0

ρ p

ρ p

0

0

ρ q 

0 1



1 0

14.12 Σ(32) The group Σ(32) contains subgroups D4 , Q4 , and Σ(8), as well as Abelian groups. In addition, it is useful to construct a discrete group as a subgroup of known groups. Here, we show one example, namely, (Z4 × Z2 )  Z2 as a subgroup of Σ(32)  (Z4 × Z4 )  Z2 . All elements of the group Σ(32) can be written in the form bk a m a n with k = 0, 1, and m, n = 0, 1, 2, 3. The generators, a, a  , and b satisfy a 4 = a 4 = b2 = e, aa  = a  a, and bab = a  . Here we define a˜ = aa  and a˜  = a 2 , where a˜ 4 = e and a˜ 2 = e. Then the elements bk a˜ m a˜ n with k, n = 0, 1, and m = 0, 1, 2, 3, generate a closed subalgebra, i.e., (Z4 × Z2 )  Z2 with ten conjugacy classes. It has eight singlets 1±0 , 1±1 , 1±2 , and 1±3 and two doublets, 21 and 22 . These conjugacy classes and characters are shown in Table 14.25. From this table, we can find decompositions of the Σ(32) representations 1±0,±1,±2,±3 , 21,0 , 23,2 , 23,0 , 22,1 , 22,0 , and 23,1 to representations of (Z4 × Z2 )  Z2 , viz., 1±0,±1,±0,±1 , 21 , 22 , 1+3 + 1−3 , and 1+2 + 1−2 , respectively. Table 14.25 Conjugacy classes and characters of (Z4 × Z2 )  Z2 h

χ±0

χ±1

χ±2

χ±3

χ 21

χ2 2

C1 :

{e},

1

1

1

1

1

2

2

C1(1) :

{a˜ a˜  },

4

1

−1

1

−1

2i

−2i

{a˜ 2 a˜ 2 },

2

1

1

1

1

−2

−2

{a˜ 3 a˜ 3 },

4

1

−1

1

−1

−2i

2i

{b, ba˜ 2 a˜ 2 },

2

±1

±1

±1

±1

0

0

{ba˜ a˜  , ba˜ 3 a˜ 3 },

4

±1

∓1

±1

∓1

0

0

{ba˜ 2 , ba˜ 2 },

4

±1

∓1

∓1

±1

0

0

{ba˜ a˜ 3 , ba˜ 3 a˜  },

2

±1

±1

∓1

∓1

0

0

{a˜ 2 , a˜ 2 },

2

1

−1

−1

1

0

0

{a˜ a˜ 3 , a˜ 3 a˜  },

2

1

1

−1

−1

0

0

C1(2) : C1(3) : C2(0) : C2(0) : C2(0) : C2(0) : C2(2,0) : (3,1) C2 :

168

14

Table 14.26 Representations of a, a  , and b in Δ(3N 2 ) for N/3 = integer

Subgroups and Decompositions of Multiplets 1k

3[k][] ⎛

a

1

a

1

b

Table 14.27 Representations of a, a  , and b in Δ(3N 2 ) for N/3 an integer

1k,

0

0

⎞

⎟ ⎜ 0 ⎠ ⎝ 0 ρk 0 0 ρ −k− ⎛ −k− ⎞ ρ 0 0 ⎜ ⎟ ρ 0 ⎠ ⎝ 0 ⎛

ωk

ρ

0

0 ⎜ ⎝0 1

1

0

0 ⎞

0

⎟ 1⎠

0

0

ρk

3[k][] ⎛

a

ω

a

ω

b

ωk

⎞ 0 ρ 0 ⎜ ⎟ 0 ⎠ ⎝ 0 ρk 0 0 ρ −k− ⎛ −k− ⎞ 0 0 ρ ⎜ ⎟ ρ 0 ⎠ ⎝ 0 0 0 ρk ⎞ ⎛ 0 1 0 ⎟ ⎜ ⎝0 0 1⎠ 1 0 0

14.13 General Δ(3N 2 ) All elements of Δ(3N 2 ) can be written in the form bk a m a n with k = 0, 1, 2, and m, n = 0, . . . , N − 1, where the generators, b, a, and a  correspond to Z3 , ZN ,  of (Z × Z  )  Z , respectively. Table 14.26 shows the different repand ZN N 3 N resentations of the generators b, a, and a  on each representation of Δ(3N 2 ) for N/3 = integer. Table 14.27 shows the same when N/3 is an integer. In general, the group Δ(3N 2 ) includes subgroups Z3 , ZN × ZN , and Δ(3M 2 ), where M is a divisor of N . In addition, when N has certain special values, it includes TN as a subgroup, as shown in Chap. 11. We consider the above decompositions in what follows. A summary of the breaking patterns is shown in Table 14.28 for N/3 integer and Table 14.29 for N/3 = integer.

14.13

General Δ(3N 2 )

Table 14.28 Breaking pattern of Δ(3N 2 ) for N/3 an integer

Table 14.29 Breaking pattern of Δ(3N 2 ) for N/3 = integer

169 Δ(3N 2 )

Z3

Δ(3N 2 ) ZN × ZN

1k,

1k

1k,

1N /3,N /3

3[k][]

10 + 11 + 12

3[k][]

1,−k− + 1k, + 1−k−,k

Δ(3N 2 )

Δ(3M 2 ) (M/3 = integer)

1k,

1k,N /M

3[k][]

3[k ][ ] (k = k  + Mn,  =  + Mn )

Δ(3N 2 )

Δ(3M 2 ) (M/3 = integer)

1k,

1k

3[k][]

3[k ][ ] (k = k  + Mn,  =  + Mn )

Δ(3N 2 ) Z3

Δ(3N 2 )

ZN × ZN

1k

1k

1k

10,0

3[k][]

10 + 11 + 12

3[k][]

1,−k− + 1k, + 1−k−,k

Δ(3N 2 ) TN

Δ(3N 2 ) Δ(3M 2 )

1k

1k

1k

1k

3[k][]

3[l−mk]

3[k][]

3[k ][ ] (k = k  + Mn,  =  + Mn )

14.13.1 Δ(3N 2 ) → Z3 The subgroup Z3 consists of elements {e, b, b2 }. There are three singlet representations 1m with m = 0, 1, 2, for Z3 , while the generator b is represented by b = ωm on 1m . When N/3 = integer, the representations 1k and 3[k][] of Δ(3N 2 ) decompose to 1k and 10 + 11 + 12 , respectively. On the other hand, when N/3 is an integer the representations 1k, and 3[k][] of Δ(3N 2 ) decompose to 1k and 10 + 11 + 12 . In both cases, the triplet components (x1 , x2 , x3 ) of Δ(3N 2 ) are decomposed to singlets of Z3 according to 10 : x1 + x2 + x3 , 11 : x1 + ω2 x2 + ωx3 , and 12 : x1 + ωx2 + ω2 x3 .

14.13.2 Δ(3N 2 ) → ZN × ZN The subgroup ZN × ZN consists of elements {a m a n } with m, n = 0, 1, . . . , N − 1. There are N 2 singlet representations 1m,n and the generators a and a  are represented by a = ρ m and a  = ρ n on 1m,n . When N/3 = integer, the representations 1k

170

14

Subgroups and Decompositions of Multiplets

and 3[k][] of Δ(3N 2 ) decompose to 10,0 and 1,−k− + 1k, + 1−k−,k , respectively. In addition, when N/3 is an integer, the representations 1k, and 3[k][] decompose to 1N /3,N/3 and 1,−k− + 1k, + 1−k−,k , respectively.

14.13.3 Δ(3N 2 ) → TN When N is any prime number except 3 or any power of such a prime number, Δ(3N 2 ) has TN as a subgroup. In the basis of Δ(3N 2 ) and TN used in the text, 2 the subgroup TN consists of all elements of the form {a˜ n bk }, where a˜ = a −m a m , n = 0, 1, . . . , N − 1, k = 0, 1, 2, and m is defined in Chap. 11 on TN . For each representation 3[k][] , we have ⎛

ρ −mk a˜ = ⎝ 0 0

0

0 0

ρ m(−mk) 0

⎞ ⎠.

(14.15)

2 ρ m (−mk)

This can be compared with the representations 3[k] , i.e., the triplets of TN : ⎛

ρk a˜ = ⎝ 0 0

⎞ 0 0 ⎠.

0 ρ km 0

ρ km

(14.16)

2

Thus the triplets 3[k][] of Δ(3N 2 ) decompose to 3[−mk] of TN . On the other hand, if  − mk = 0 (mod N ), the triplets decompose to singlets 10 + 11 + 12 of TN and their components (x1 , x2 , x3 ) correspond to 1 0 : x 1 + x2 + x 3 ,

11 : x1 + ω2 x2 + ωx3 ,

12 : x1 + ωx2 + ω2 x3 . (14.17)

For instance, when N = 7, there are 16 triplets in Δ(147) and two triplets in T7 . Triplets 3[0][1] , 3[0][2] , 3[0][4] , 3[1][3] , 3[1][4] , 3[2][6] , and 3[4][5] decompose to 31 of T7 . In the same way, triplets 3[0][3] , 3[0][5] , 3[0][6] , 3[1][1] , 3[2][2] , 3[3][5] , and 3[4][4] decompose to 32 of T7 . The remaining triplets, i.e., 3[1][2] and 3[3][6] , decompose to 10 + 11 + 12 of T7 . As far as singlets are concerned, since the representations are a = 1, a  = 1, a˜ = 1, and b = ωn in both Δ(3N 2 ) and TN , their singlets correspond.

14.13.4 Δ(3N 2 ) → Δ(3M 2 ) We consider the subgroup Δ(3M 2 ), where M is a divisor of N . We define a˜ = a p and a˜  = a p , with p = N/M, whence p is an integer. The subgroup Δ(3M 2 ) consists of all elements of the form bk a˜ m a˜ n with k = 0, 1, 2, and m, n = 0, . . . , M − 1.

14.13

General Δ(3N 2 )

171

Table 14.30 Representations of a, ˜ a˜  , and b˜ in Δ(3N 2 ) for N/3 = integer

1k,

3[k][] ⎛

a˜

ωp

a˜ 

ωp

ρ p

Table 14.31 Representations of a, ˜ a˜  , and b˜ in Δ(3N 2 ) for N/3 = integer

0

0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

1k

3[k][] ⎛

1

a˜ 

1

ωk

ρ p

ρ pk

0

0

⎞

0

⎟ ⎜ ρ pk 0 ⎠ ⎝ 0 −p(k+) 0 0 ρ ⎛ −p(k+) ⎞ ρ 0 0 ⎜ ⎟ 0 ⎠ 0 ρ p ⎝ ⎛

b

⎞

ωk

a˜

⎞

0

⎜ ⎟ 0 ρ pk ⎝ 0 ⎠ −p(k+) 0 0 ρ ⎛ −p(k+) ⎞ 0 0 ρ ⎜ ⎟ 0 ρ p 0 ⎠ ⎝ ⎛

b

0

0

⎞

0

ρ pk

0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

Table 14.30 shows the various representations of a, ˜ a˜  , and b˜ on each representation 2 of Δ(3N ) for N/3 integer. In addition, Table 14.31 shows the representations of a, ˜ a˜  , and b˜ on each representation of Δ(3N 2 ) for N/3 = integer. There are three types of combination (N, M), i.e., (1) both N/3 and M/3 are integers, (2) N/3 is an integer, but M/3 is not an integer, and (3) neither N/3 nor M/3 is an integer. When both N/3 and M/3 are integers, the representations 1k, and 3[k+Mn][+Mn ] of Δ(3N 2 ) decompose to representations 1k,p and 3[k][] of Δ(3M 2 ), where n and n are integers. Next we consider the case where N/3 is an integer and M/3 = integer, where p = N/M must be 3n. In this case, the representations 1k, and 3[k+Mn][+Mn ] of Δ(3N 2 ) decompose to representations 1k and 3[k][] of Δ(3M 2 ), respectively. The last case is when neither N/3 nor M/3 is an integer. In this case, the representations 1k and 3[k+Mn][+Mn ] of Δ(3N 2 ) decompose to representations 1k and 3[k][] of Δ(3M 2 ).

172

14

Table 14.32 Representations of a, a  , and b in Δ(27)

Subgroups and Decompositions of Multiplets 1k,

3[k][] ⎛

a

ω

a

ω

Table 14.33 Breaking pattern of Δ(27)

0

0

⎞

⎜ ⎟ 0 ⎠ ⎝ 0 ωk 0 0 ω−k− ⎛ −k− ⎞ 0 0 ω ⎜ ⎟ ω 0 ⎠ ⎝ 0 ⎛

0

0 ⎞

ωk

0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

ωk

b

ω

Δ(27)

Z3

Δ(27)

Z3 × Z3

1k,

1k

1k,

1,

3[k][]

10 + 11 + 12

3[k][]

1,−k− + 1k, + 1−k−,k

14.14 Δ(27) All elements of the group Δ(27) can be expressed in the form bk a m a n with k = 0, 1, 2, and m, n = 0, 1, 2, where the generators b, a, and a  correspond to the Z3 groups of (Z3 × Z3 )  Z3 . Table 14.32 shows the representations of the generators b, a, and a  for each representation of Δ(27). As subgroups, the group Δ(27) contains Abelian subgroups Z3 and Z3 × Z3 . The breaking pattern of Δ(27) is summarized in Table 14.33.

14.14.1 Δ(27) → Z3 The subgroup Z3 consists of the elements {e, b, b2 }. There are three singlet representations 1m with m = 0, 1, 2, for Z3 and the generator b is represented by b = ωm on 1m . When N/3 = integer, the representations 1k, and 3[k][] of Δ(27) decompose to 1k and 10 + 11 + 12 , respectively. The triplet components (x1 , x2 , x3 ) decompose to singlets 10 : x1 + x2 + x3 , 11 : x1 + ω2 x2 + ωx3 , and 12 : x1 + ωx2 + ω2 x3 of Z3 .

14.14.2 Δ(27) → Z3 × Z3 The subgroup Z3 × Z3 consists of elements of the form {a m a n } with m, n = 0, 1, 2. There are 9 singlet representations 1m,n and the generators a and a  are represented

14.15

General TN

Table 14.34 Breaking pattern of TN

173 TN

Z3

TN

ZN

1k

1k

1k

10

3[k] 3¯ [k]

10 + 11 + 12

3[k] 3¯ [k]

1k + 1km + 1km2

10 + 11 + 12

1−k + 1−km + 1−km2

by a = ωm and a  = ωn on 1m,n . The representations 1k, and 3[k][] of Δ(27) decompose to 1, and 1,−k− + 1k, + 1−k−,k , respectively.

14.15 General TN Since the group TN is isomorphic to ZN  Z3 , ZN and Z3 appear as subgroups. Recall that all elements of TN can be written in the form a m bk with m = 0, . . . , N − 1, and k = 0, 1, 2. There are three singlets 1k and 2(N − 1)/6 triplets 3[k  ] and 3¯ [k  ] with k  = 1, . . . , (N − 1)/6. On the triplets 3[k] , the generators a and b are represented by ⎞ ⎛ ⎞ ⎛ k 0 0 ρ 0 1 0 0 ⎠, (14.18) b = ⎝0 0 1⎠, a = ⎝ 0 ρ km 2 km 1 0 0 0 0 ρ where ρ = e2πi/N . For each N , we have the value of m, as explained in Chap. 11 on TN . The representations of 3¯ [k] can be obtained by changing ρ to ρ −1 . In the following, we obtain in detail the general breaking patterns of TN → Z3 and TN → ZN . A summary of the breaking patterns is shown in Table 14.34.

14.15.1 TN → Z3 The two elements e and b generate the subgroup Z3 . Obviously, there are three singlet representations 10 , 11 , and 12 . That is, we have b = 1 on 10 , b = ω on 11 , and b = ω2 on 12 . The singlets 1k of TN become 1k of Z3 . The triplets 3[k] of TN , viz., (xk , xkm , xkm2 ), decompose to three singlets 10 : xk + xkm + xkm2 ,

11 : xk + ω2 xkm + ωxkm2 ,

12 : xk + ωxkm + ω2 xkm2 .

Decompositions of triplets 3¯ [k] are the same.

14.15.2 TN → ZN The subgroup ZN comprises the elements {e, a, . . . , a N−1 }. It is a normal subgroup of TN and there are N types of irreducible singlet representation 10 , 11 , . . . , 1N−1 .

174

14

Table 14.35 Representations of a and b in T7

10

11

Subgroups and Decompositions of Multiplets

12

⎛ a

1

1

1

b

1

ω

ω2

3¯

3 ρ

0

0

⎟ ⎜ ⎝ 0 ρ2 0 ⎠ 0 0 ρ4 ⎛ ⎞ 0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

Table 14.36 Breaking pattern of T7

⎞

⎛

ρ −1

0

0

⎞

⎜ ⎟ 0 ⎠ ρ −2 ⎝ 0 0 0 ρ −4 ⎛ ⎞ 0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

T7

Z3

T7

1k

1k

1k

10

3 3¯

10 + 11 + 12

3 3¯

11 + 12 + 14

10 + 11 + 12

Z7

13 + 15 + 16

On the singlets 1k of TN , the generator a is represented by a = 1. Therefore the singlets 1k of TN become 10 of ZN . The triplets 3[k] : (xk , xkm , xkm2 ) decompose to three singlets 1k , 1km , and 1km2 . The triplets 3¯ [k] decompose to 1−k , 1−km , and 1−km2 .

14.16 T7 All elements of the group T7 can be expressed in the form bm a n with m = 0, 1, 2, and n = 0, . . . , 6, where b3 = e and a 7 = e. Table 14.35 shows the various representations of generators a and b for each representation of T7 . As subgroups, T7 contains Abelian subgroups Z3 and Z7 . The summary of the breaking pattern is shown in Table 14.36.

14.16.1 T7 → Z3 The subgroup Z3 consists of elements {e, b, b2 }. The three singlet representations 1m of Z3 with m = 0, 1, 2, are specified in such a way that b = ωm on 1m . Then the representations 10 , 11 , 12 , 3, and 3¯ of T7 decompose to 10 , 11 , 12 , 10 + 11 + 12 , and 10 + 11 + 12 , respectively. Here the T7 triplets 3 : (x1 , x2 , x4 ) and 3¯ : (x6 , x5 , x3 ) decompose to three singlets, namely, 10 + 11 + 12 , and their components correspond to 1 0 : x1 + x 2 + x 4 ,

11 : x1 + ω2 x2 + ωx4 ,

12 : x1 + ωx2 + ω2 x4 ,

10 : x 6 + x 5 + x 3 ,

11 : x6 + ω2 x5 + ωx3 ,

12 : x6 + ωx5 + ω2 x3 .

14.17

General Σ(3N 3 )

175

Table 14.37 Breaking pattern of Σ(3N 3 ) Σ(3N 3 ) Z3

Σ(3N 3 ) ZN × ZN × ZN

Σ(3N 3 ) Δ(3N 2 ) (N/3 = integer)

1k,

1k,

1k,

1k

3[k][][m] 10 + 11 + 12

1,,

3[k][][m] 1k,,m + 1m,k, + 1,m,k

1k

3[k][][m] 3[−k][k−m]

Σ(3N 3 )

Δ(3N 2 ) (N/3 = integer)

Σ(3N 3 )

Σ(3M 2 )

1k,

1k,

1k,+Mn

1k,

3[k][][m]

3[−k][k−m]

3[k+Mn][+Mn ][m+Mn ]

3[k][][m]

14.16.2 T7 → Z7 The subgroup Z7 consists of elements a n with n = 0, . . . , 6. The seven singlets 1m of Z7 with m = 0, . . . , 6 are specified in such a way that b = ρ m on 1m , where ρ = e2πi/7 . Then the representations 10 , 11 , 12 , 3, and 3¯ of T7 decompose to 10 , 10 , 10 , 11 + 12 + 14 , and 13 + 15 + 16 , respectively.

14.17 General Σ(3N 3 ) Recall that all elements of the group Σ(3N 3 ) can be written in the form bk a  a m a n with k = 0, 1, 2, and , m, n = 0, 1, . . . , N − 1. As subgroups, the group Σ(3N 3 ) contains Z3 , ZN × ZN × ZN , Δ(3N 2 ), and Σ(3M 3 ), where M is a divisor of N . The summary of the breaking pattern of Σ(3N 3 ) is shown in Table 14.37.

14.17.1 Σ(3N 2 ) → ZN × ZN × ZN The subgroup ZN × ZN × ZN consists of elements a k a  a m with k, , m = 0, . . . , N − 1. Obviously, it is a normal subgroup of Σ(3N 3 ). There are N 3 singlet representations 1k,,m and the generators a, a  , and a  are represented by a = ρ k , a  = ρ  , and a  = ρ m on 1k,,m . Then the representations 1k, and 3[k][][m] of Σ(3N 3 ) decompose to 1,, and 1k,,m + 1m,k, + 1,m,k , respectively.

14.17.2 Σ(3N 3 ) → Δ(3N 2 ) The subgroup Δ(3N 2 ) consists of the elements a˜ k a˜  bm with k,  = 0, . . . , N − 1, and m = 0, 1, 2. There are 3 singlet representations 1k and (N 2 − 1)/3 triplet representations 3[k][] for N/3 = integer, and 9 singlet representations 1k, and

176

14

Subgroups and Decompositions of Multiplets

Table 14.38 Representations of a, a  , a  , and b in Σ(81) 1k,

3A

3B

⎛ a

ω

a

ω

ω 0 ⎜ ⎝0 1 0 0 ⎛ 1 0 ⎜ ⎝0 ω ⎛

a 

b

ω

ωk

0

0

1 0

⎜ ⎝0 0 ⎛ 0 ⎜ ⎝0

1 0 1 0

1 0

0

⎞

⎟ 0⎠ 1 ⎞ 0 ⎟ 0⎠ 1 ⎞ 0 ⎟ 0⎠ ω ⎞ 0 ⎟ 1⎠ 0

3C

⎛

ω2 0 ⎜ ⎝ 0 ω 0 0 ⎛ ω 0 ⎜ ⎝ 0 ω2 ⎛

0

0

ω

0

⎜ ⎝0 ω 0 0 ⎛ 0 1 ⎜ ⎝0 0 1 0

0

⎞

⎟ 0⎠ ω ⎞ 0 ⎟ 0⎠ ω ⎞ 0 ⎟ 0 ⎠ ω2 ⎞ 0 ⎟ 1⎠ 0

3D

⎛

1 0 ⎜ ⎝ 0 ω2 0 0 ⎛ 2 ω 0 ⎜ ⎝ 0 1 ⎛

0 ω2

⎜ ⎝ 0 0 ⎛ 0 ⎜ ⎝0 1

0

⎞

⎟ 0 ⎠ ω2 ⎞ 0 ⎟ 0 ⎠ 0 ω2 ⎞ 0 0 ⎟ ω2 0 ⎠ 0 1 ⎞ 1 0 ⎟ 0 1⎠ 0 0

⎞ ω2 0 0 ⎟ ⎜ ⎝ 0 1 0⎠ 0 0 ω ⎛ ⎞ ω 0 0 ⎜ ⎟ ⎝ 0 ω2 0 ⎠ 0 0 1 ⎞ ⎛ 1 0 0 ⎟ ⎜ ⎝0 ω 0 ⎠ 0 0 ω2 ⎛ ⎞ 0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0 ⎛

(N 2 − 3)/3 triplet representations 3[k][] for N/3 integer. The generators a˜ and a˜  are represented by a˜ = aa −1 and a˜ = a  a −1 . Then the representations 1k, and 3[k][][m] of Σ(3N 3 ) decompose to 1k and 3[−k][k−m] for N/3 = integer and 1k, and 3[−k][k−m] for N/3 integer.

14.17.3 Σ(3N 3 ) → Σ(3M 3 ) We consider the subgroup Σ(3M 3 ), where M is a divisor of N . We denote a˜ = a p , a˜  = a p , and a˜  = a p , with p = N/M, whence p is an integer. The subgroup Σ(3M 3 ) consists of all elements of the form bk a˜  a˜ m a˜ n with k = 0, 1, 2, and , m, n = 0, . . . , M − 1. The singlets 1k,+Mn and triplets 3[k+Mn][+Mn ][m+Mn ] of Σ(3N 3 ) correspond to singlets 1k, and triplets 3[k][][m] of Σ(3M 3 ), where n, n , n are integers.

14.18 Σ(81) All elements of the group Σ(81) can be written in the form bk a  a m a n with k, , m, n = 0, 1, 2, where these generators satisfy a 3 = a 3 = a 3 = 1, aa  = a  a, aa  = a  a, a  a  = a  a  , b3 = 1, b2 ab = a  , b2 a  b = a, and b2 a  b = a  . Table 14.38 shows the representations of generators b, a, a  , and a  for each representation of Σ(81). As subgroups, the group Σ(81) contains Z3 × Z3 × Z3 and Δ(27). The breaking pattern of Σ(81) is summarized in Table 14.39.

14.18

177

Σ(81)

Table 14.39 Breaking pattern of Σ(81)

Σ(81)

Z3 × Z3 × Z3

Σ(81)

Δ(27)

1k,

1,,

1k,

1k,0

3A

10,0,0 + 10,1,0 + 10,0,1

3A

3[0][1]

3B

12,1,1 + 11,2,1 + 11,1,2

3B

3[0][1]

3C

10,2,2 + 12,0,2 + 12,2,0

3C

3[0][1]

3D 3¯ A

12,1,0 + 10,2,1 + 11,0,2

3D 3¯ A

10,2 + 11,2 + 12,2

3¯ B 3¯ C

11,2,2 + 12,1,2 + 12,2,1

3¯ B 3¯ C

3[0][2]

12,0,0 + 10,2,0 + 10,0,2 10,1,1 + 11,0,1 + 11,1,0

3¯ D

11,2,0 + 10,1,2 + 12,0,1

3¯ D

3[0][2] 3[0][2] 10,1 + 11,1 + 12,1

14.18.1 Σ(81) → Z3 × Z3 × Z3 The subgroup Z3 × Z3 × Z3 consists of the elements {e, a, a 2 , a  , a 2 , a  , a 2 , . . .}. There are 33 singlets 1k,,m of Z3 × Z3 × Z3 and the generators a, a  , and a  are represented on 1k,,m by a = ωk , a  = ω , and a  = ωm . Then the singlet 1k, of Σ(81) decomposes to 1,, . Regarding the triplets, they become three different singlets of Z3 × Z3 × Z3 , as shown in detail in Table 14.39.

14.18.2 Σ(81) → Δ(27) The subgroup Δ(27) consists of elements of the form bk a˜ m a˜ n , where a˜ = a 2 a  and ˜ and a˜  for a˜  = a  a 2 . Table 14.40 shows the representations of the generators b, a, each representation of Σ(81). Then the singlet 1k, decomposes to 1k,0 of Δ(27). Triplets 3A,B,C and 3¯ A,B,C become triplets 3[0][1] and 3[0][2] , respectively, while 3D Table 14.40 Representations of b, a, ˜ and a˜  of Δ(27) in Σ(81) 1k,

a˜

a˜ 

b

1

1

ωk

3A

3B

3C

3D

⎛

⎛

⎛

⎛

⎞ ω 0 0 ⎜ ⎟ ⎝0 1 0 ⎠ 0 0 ω2 ⎞ ⎛ 2 0 0 ω ⎟ ⎜ ⎝ 0 ω 0⎠ 0 0 1 ⎛ ⎞ 0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

⎞ ω 0 0 ⎜ ⎟ ⎝0 1 0 ⎠ 0 0 ω2 ⎛ 2 ⎞ ω 0 0 ⎜ ⎟ ⎝ 0 ω 0⎠ 0 0 1 ⎛ ⎞ 0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

⎞ ω 0 0 ⎜ ⎟ ⎝0 1 0 ⎠ 0 0 ω2 ⎛ 2 ⎞ ω 0 0 ⎜ ⎟ ⎝ 0 ω 0⎠ 0 0 1 ⎛ ⎞ 0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

ω2 ⎜ ⎝ 0 0 ⎛ 2 ω ⎜ ⎝ 0 ⎛

0 ω2 0 0 ω2

0 0

⎜ ⎝0 1

0

0 1

0

⎞

⎟ 0 ⎠ ω2 ⎞ 0 ⎟ 0 ⎠ ω2 ⎞

0

⎟ 1⎠

0

0

178 Table 14.41 Breaking pattern of Δ(6N 2 ) for N/3 = integer

14

Subgroups and Decompositions of Multiplets

Δ(6N 2 )

Σ(2N 2 )

10

1+0

11

1−0

2

1+0 + 1−0

31k

1+k + 20,−k

32k

1−k + 20,−k

6[[k],[]]

2−k−,− + 2,−k + 2k,k+

Δ(6N 2 )

Δ(3N 2 )

10

10

11

10

2

11 + 12

31k

3[−k][k]

32k

3[−k][k]

6[[k],[]]

3[k][] + 3[−][k+]

Δ(6N 2 )

Δ(6M 2 ) (M/3 = integer)

10

10

11

11

2

2

31(k+Mn)

31k

32(k+Mn)

32k

6[[k+Mn],[+Mn ]]

6[[k],[]]

and 3¯ D each correspond to three singlets, viz., 10,2 + 11,2 + 12,2 and 10,1 + 12,1 + 11,1 , respectively.

14.19 General Δ(6N 2 ) All elements of Δ(6N 2 ) can be written in the form bk c a m a n with k = 0, 1, 2,  = 0, 1, and m, n = 0, . . . , N − 1, where the generators b, c, a, and a  correspond  of (Z × Z  )  S , respectively. We now examine the to Z3 , Z2 , ZN , and ZN N 3 N different possible breaking patterns. In particular, Δ(6N 2 ) can break into Σ(2N 2 ) and Δ(3N 2 ). The breaking pattern of Δ(6N 2 ) is summarized in Table 14.41 for N/3 = integer and Table 14.42 for N/3 integer. For further breaking of these two groups, see Sect. 14.11 for Σ(2N 2 ) and Sect. 14.13 for Δ(3N 2 ).

14.19

General Δ(6N 2 )

179

Table 14.42 Breaking pattern of Δ(6N 2 ) for N/3 an integer Δ(6N 2 )

Σ(2N 2 )

Δ(6N 2 )

Δ(3N 2 )

10

1+0

10

10,0

11

1−0

11

10,0

21

1+0 + 1−0

21

11,0 + 12,0

22

22N/3,N/3

22

11,2 + 12,1

23

2N/3,2N/3

23

11,1 + 12,2

24

2N/3,2N/3

24

10,1 + 10,2

31k

1+k + 20,−k

31k

3[−k][k]

32k

1−k + 20,−k

32k

3[−k][k]

6[[k],[]]

2−k−,− + 2,−k + 2k,k+

6[[k],[]]

3[k][] + 3[−][k+]

Δ(6N 2 )

Δ(6M 2 ) (M/3 = integer)

Δ(6N 2 )

Δ(6M 2 ) (M/3 = integer)

10

10

10

10

11

11

11

11

21

2

21

21

22

2

22

22

23

2

23

23

24

10 + 11

24

24

31(k+Mn)

31k

31(k+Mn)

31k

32(k+Mn)

32k

32(k+Mn)

32k

6[[k+Mn],[+Mn ]]

6[[k],[]]

6[[k+Mn],[+Mn ]]

6[[k],[]]

Table 14.43 Representations of Σ(2N 2 )

1+n

1−n

ρn

ρn

2p,q 

a −1

 a

ρn

ρn 

b

1

−1

ρq

0

0

ρp

ρp

0

0

ρq 

0

1

1

0

 

14.19.1 Δ(6N 2 ) → Σ(2N 2 ) The subgroup Σ(2N 2 ) consists of elements of the form {c a −m a n }. There are 2N singlets and N(N − 1)/2 doublets for Σ(2N 2 ). The representations of the generators are summarized in Table 14.43. When N/3 = integer, the singlets 10 and 11 correspond to 1+0 and 1−0 , the doublet 2 becomes 1+0 + 1−0 , and the triplets

180

14

Table 14.44 Representations of Δ(3N 2 ) for 3N = integer (left) and 3N integer (right)

1k

3[k][] ⎛

a

1

a

1

b

ρ

1k, 3[k][] 0

0

⎞

⎟ ⎜ 0 ⎠ ⎝ 0 ρk 0 0 ρ −k− ⎛ −k− ⎞ ρ 0 0 ⎜ ⎟ ρ 0 ⎠ ⎝ 0 ⎛

ωk

Subgroups and Decompositions of Multiplets

0

0 ⎞

0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

⎛ a

ω

a  ω

ρk b

0

0

⎞

⎜ ⎟ 0 ⎠ ⎝ 0 ρk 0 0 ρ −k− ⎛ −k− ⎞ 0 0 ρ ⎜ ⎟ ρ 0 ⎠ ⎝ 0 ⎛

ωk

ρ

0

0 ⎞

ρk

0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0

31k and 32k decompose to 1+k + 20,−k and 1−k + 20,−k , respectively. The sextet 6[[k],[]] decomposes to 2−k−,− + 2,−k + 2k,k+ . The decomposition of 32k , written (x1 , x2 , x3 ), is non-trivial. It decomposes to (x2 )1−k + (x1 , −x3 )20,−k . Similarly, when N/3 is an integer, the singlets 10 and 11 correspond to 1+0 and 1−0 , and doublets 21 , 22 , 23 , and 24 become 1+0 + 1−0 , 22N/3,N/3 , 2N/3,2N/3 , and 2N/3,2N/3 , respectively. The triplets 31k and 32k decompose to 1+k + 20,−k and 1−k + 20,−k , while the sextet 6[[k],[]] decomposes to 2−k−,− + 2,−k + 2k,k+ .

14.19.2 Δ(6N 2 ) → Δ(3N 2 ) The subgroup Δ(3N 2 ) consists of elements of the form {bk a m a n }. The representations of the generators are summarized in Table 14.44. When N/3 = integer, the singlets 10 and 11 correspond to 10 and 10 , the doublet 2 becomes 11 + 12 , and the triplets 31k and 32k correspond to 3[−k][k] and 3[−k][k] , respectively. The sextet 6[[k],[]] decomposes to 3[k][] + 3[][k+] . When N/3 is an integer, the singlets 10 and 11 correspond to 10,0 and 10,0 , and the doublets 21 , 22 , 23 , and 24 become 11,0 + 12,0 , 11,2 + 12,1 , 11,1 + 12,2 , and 10,1 + 10,2 , respectively. The triplets 31k and 32k correspond to 3[−k][k] and 3[−k][k] , while the sextet 6[[k],[]] decomposes to 3[k][] + 3[][k+] .

14.19.3 Δ(6N 2 ) → Δ(6M 2 ) We consider the subgroup Δ(6M 2 ), where M is a divisor of N . We define a˜ = a p and a˜  = a p with p = N/M, whence p is an integer. The subgroup Δ(6M 2 ) consists of elements of the form bk c˜ a m a˜ n with k = 0, 1, 2,  = 0, 1, and m, n = 0, . . . , M − 1. There are three types of combination (N, M), i.e., (1) both N/3 and M/3 are integers, (2) N/3 is an integer, but M/3 is not an integer, and (3) neither N/3 nor M/3 is an integer.

14.20

181

Δ(54)

Table 14.45 Representations of a, a  , b, and c in Δ(54) 10 11 21  a 1

1

a 1

1





1 0





0 1 

1 

c 1 −1

1 0 0 1



b 1

22

ω

0

0

ω2

0 1



1 0

 



23

ω2

0

0

ω

ω2

0

0

ω

ω

0

0

ω2

0

1

1

0



 

24

ω 0

 

ω 0

 

ω 0



0 1

31k

⎞ 0 0 ωk ⎟ ⎜ ⎝ 0 ω2k 0 ⎠ ω2 0 ω2 0 0 1 ⎛ ⎞    1 0 0 ω 0 0 ⎜ ⎟ 0 ⎠ ⎝ 0 ωk 0 ω2 ω2 0 0 ω2k ⎛ ⎞    0 1 0 0 1 0 ⎜ ⎟ ⎝0 0 1⎠ ω2 0 1 1 0 0 ⎛ ⎞    0 0 1 0 1 1 ⎜ ⎟ ⎝0 1 0⎠ 0 1 0 1 0 0 0

 

ω

0



⎛

32k ⎛

⎞ 0 0 ωk ⎜ ⎟ ⎝ 0 ω2k 0 ⎠ 0 0 1 ⎛ ⎞ 1 0 0 ⎟ ⎜ 0 ⎠ ⎝ 0 ωk 0 0 ω2k ⎛ ⎞ 0 1 0 ⎜ ⎟ ⎝0 0 1⎠ 1 0 0 ⎛ ⎞ 0 0 −1 ⎜ ⎟ ⎝ 0 −1 0 ⎠ −1 0 0

When both N/3 and M/3 are integers, each representation of Δ(6N 2 ) decomposes to representations of Δ(6M 2 ) as follows: the singlets 10 and 11 and the doublet 2k remain the same, while the triplets 31(k+Mn) and 32(k+Mn) decompose to 31k and 32k , and the sextet 6[[k+Mn],[+Mn ]] corresponds to 6[[k],[]] , where n and n are integers. Next we consider the case when N/3 integer and M/3 = integer, where p = N/M must be 3n. In this case, the singlets 10 and 11 remain the same representations, while the doublets 21 , 22 , and 23 correspond to 2, and 24 decomposes to 10 + 11 because p has a factor of three. The triplets 31(k+Mn) and 32(k+Mn) correspond to 31k and 32k , and the sextet 6[[k+Mn],[+Mn ]] corresponds to 6[[k],[]] . The last case is when neither N/3 nor M/3 is an integer. Then the singlets 10 and 11 and the doublet 2 remain the same. The triplets 31(k+Mn) and 32(k+Mn) decompose to 31k and 32k , while the sextet 6[[k+Mn],[+Mn ]] corresponds to 6[[k],[]] .

14.20 Δ(54) All elements of the group Δ(54) can be expressed in the form bk c a m a n with k, m, n = 0, 1, 2, and  = 0, 1. Here, the generators a and a  correspond to Z3 and Z3 of (Z3 × Z3 )  S3 , respectively, while b and c correspond to Z3 and Z2 in S3 of (Z3 × Z3 )  S3 , respectively. Table 14.45 shows the representations of the generators b, c, a, and a  for each representation of Δ(54). Regarding subgroups, the group Δ(54) contains S3 × Z3 , Σ(18), and Δ(27). The breaking pattern of Δ(54) is summarized in Table 14.46.

182 Table 14.46 Breaking pattern of Δ(54)

14

Subgroups and Decompositions of Multiplets

Δ(54)

S3 × Z3

Δ(54) Σ(18)

Δ(54)

Δ(27)

10

10

10

1+0

10

10,0

11

10

11

1−0

11

10,0

21

20

21

20,0

21

11,0 + 12,0

22

20

22

22,1

22

11,1 + 12,2

23

20

23

21,2

23

11,2 + 12,1

24

10 + 10

24

21,2

24

10,1 + 10,2

31k

1k + 2k

31k

1+k + 20,2k

31k

3[0][k]

32k

1k

32k

1−k + 22k,0

32k

3[0][k]

Table 14.47 Representations of b and c of S3 in Δ(54)

+ 2k

1

1

2 

b

1

1 

c

1

−1

ω

0

0 0

ω2  1

1

0



14.20.1 Δ(54) → S3 × Z3 The group Δ(54) contains S3 × Z3 as a subgroup. The subgroup S3 consists of elements {e, b, c, b2 , bc, b2 c}. The Z3 part of S3 × Z3 consists of elements {e, aa 2 , a 2 a  }, where (aa 2 )3 = e and the element aa 2 commutes with all the S3 elements. Representations r k for S3 × Z3 are specified by representations r of S3 and the Z3 charge k, where r = 1, 1 , 2 and k = 0, 1, 2. That is, the element aa 2 is represented by aa 2 = ωk on r k for k = 0, 1, 2. For the decomposition of Δ(54) to S3 × Z3 , it is convenient to use the basis for S3 representations 1, 1 , and 2, which is shown in Table 14.47. Then the representations 10 , 11 , 21 , 22 , 23 , 24 , 31(k) , and 32(k) of Δ(54) decompose to representations 10 , 10 , 20 , 20 , 20 , 10 + 10 , 1k + 2k , and 1k + 2k of S3 × Z3 , for k = 1, 2. Components of S3 doublets and singlets obtained from Δ(54) triplets are the same as those considered in the decomposition for S4 → S3 .

14.20.2 Δ(54) → Σ(18) We consider the subgroup Σ(18), which consists of elements b˜  a˜ m a n with  = 0, 1, and m, n = 0, 1, 2, where b˜ = c and a˜ = a 2 . Table 14.48 shows the representations of the generators a, ˜ a  , and b˜ for each representation of Δ(54). Then the representations 10 , 11 , 21 , 22 , 23 , 24 , 31k , and 32k of Δ(54) decompose to representations

14.20

183

Δ(54)

˜ a, Table 14.48 Representations of b, ˜ and a  of Σ(18) in Δ(54) 10 11 21  a˜ 1

1 

a

1

1 

b˜ 1 −1

22

1

0

0

1

1

0

0

1

0

1

1

0

 

 

 

23

ω

0

0

ω2

ω2

0

0

ω

0 1



 

 



1 0

24

ω2

0

0

ω

ω

0

0

ω2

0 1 1 0



 

31k

ω2 0

 

ω 0



0 1



⎛

ω2k

32k 0

0

⎞ ⎛

⎜ ⎟ ⎝ 0 ωk 0 ⎠ 0 0 1 ⎞ ⎛  1 0 0 0 ⎟ ⎜ 0 ⎠ ⎝ 0 ωk ω2 0 0 ω2k ⎛ ⎞  0 0 1 1 ⎜ ⎟ ⎝0 1 0⎠ 0 1 0 0 0

ω

ω2k

0

0

⎞

⎜ ⎟ ⎝ 0 ωk 0 ⎠ 0 0 1 ⎛ ⎞ 1 0 0 ⎜ ⎟ 0 ⎠ ⎝ 0 ωk 0 0 ω2k ⎛ ⎞ 0 0 −1 ⎜ ⎟ ⎝ 0 −1 0 ⎠ −1 0 0

1+0 , 1−0 , 20,0 , 22,1 , 21,2 , 21,2 , 1+k + 20,2k , and 1−k + 22k,0 of Σ(18), respectively. The decomposition of triplet components is obtained as follows: ⎛ ⎞ ⎛ ⎞     x1 x1 x x3 ⎝ ⎠ ⎝ x2 ⎠ → (x2 )1+k ⊕ 1 x2 , → (x2 )1−k ⊕ . x3 2 −x1 2 0,2k 2k,0 x3 3 x3 3 1k 2k (14.19)

14.20.3 Δ(54) → Δ(27) We consider the subgroup Δ(27), which consists of elements of the form bk a m a n , with k, m, n = 0, 1, 2. Using Table 14.45, it is found that the representations 10 , 11 , 21 , 22 , 23 , 24 , 31k , and 32k of Δ(54) decompose to representations 10,0 , 10,0 , 11,0 + 12,0 , 11,1 + 12,2 , 11,2 + 12,1 , 10,2 + 10,1 , 3[0][k] , and 3[0][k] of Δ(27), respectively.

Chapter 15

Anomalies

15.1 Generic Aspects Several interesting applications of Abelian and non-Abelian discrete symmetries have been studied in areas of particle physics such as flavor physics (see Chap. 16). In general, symmetries at the tree level can be broken by quantum effects, that is, anomalies. When symmetries are anomalous, symmetry-breaking terms are induced. On the other hand, symmetries should be anomaly-free to be exact, even including any quantum effects. Anomalies of continuous symmetries, in particular gauge symmetries, have been well studied. However, anomalies of non-Abelian discrete symmetries, or indeed Abelian ones, may not be so well-known. Here we review anomalies of Abelian and non-Abelian discrete symmetries. We study the gauge theory with a (non-Abelian) gauge group Gg and a set of fermions Ψ = [ψ (1) , . . . , ψ (M) ]. We assume that their Lagrangian is invariant under the following chiral transformation: Ψ (x) → U Ψ (x),

(15.1)

where U = exp(iαPL ) and α = α A TA , with TA the generators of the transformation, α A the transformation parameters, and PL the left-chiral projector. The above transformation is not necessarily a gauge transformation. The fermions Ψ (x) carry the (irreducible) M-plet representation R M . For the moment, we consider an Abelian flavor symmetry and we suppose that Ψ (x) corresponds to (non-trivial) singlets under the flavor symmetry, while they correspond to the R M representation under the gauge group Gg . Since the generators TA and also α are represented on R M as an (M × M) matrix, we use the notation TA (R M ) and α(R M ) = α A TA (R M ). For our purposes, the path integral approach is convenient. Thus, we use Fujikawa’s method [1, 2] to derive anomalies of continuous and discrete symmetries (see, e.g., [3]). We calculate the transformation of the path integral measure: DΨ DΨ¯ → DΨ DΨ¯ J (α), H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_15, © Springer-Verlag Berlin Heidelberg 2012

(15.2) 185

186

15

where the Jacobian J (α) is written as    4 J (α) = exp i d xA(x; α) .

Anomalies

(15.3)

The anomaly function A consists of a gauge part and a gravitational part [4–6]: A = Agauge + Agrav .

(15.4)

The gauge part is given by Agauge (x; α) =

   1 μν (x) , Tr α R M F μν (x)F 2 32π

(15.5)

where F μν denotes the field strength of the gauge fields given by Fμν = [Dμ , Dν ], μν denotes its dual given by F μν = εμνρσ Fρσ . The trace Tr runs over all and F internal indices. When the transformation corresponds to a continuous symmetry, this anomaly can be calculated by the triangle diagram with external lines of two gauge bosons and one current corresponding to the symmetry for (15.1). Similarly, the gravitational part is obtained by [4–6] Weyl fermion

Agrav = −Agrav

   tr α R (M) ,

(15.6)

where tr denotes the trace for the (M × M) matrix TA (R M ). The contribution of a single Weyl fermion to the gravitational anomaly is given by [4–6] Weyl fermion

Agrav

=

1 1 μνρσ ε Rμν λγ Rρσ λγ . 384π 2 2

(15.7)

When other sets of Mi -plet fermions ΨMi are included in a

theory, the total gauge and gravity anomalies are obtained by summing to give ΨM Agauge and i

ΨMi Agrav . We evaluate these anomalies by using the following index theorems [4, 5]:  1 μνρσ a b d4 x ε Fμν Fρσ tr[ta tb ] ∈ Z, (15.8a) 32π 2  1 1 μνρσ 1 Rμν λγ Rρσ λγ ∈ Z, (15.8b) ε d4 x 2 384π 2 2 where ta are generators of Gg in the fundamental representation. We use the convention that tr[ta tb ] = δab /2. The factor 1/2 in (15.8b) follows from Rohlin’s theorem [7], as discussed in [8]. Of course, these indices are independent of each other. The path integral includes all possible configurations corresponding to different index numbers. First of all, we study anomalies of the continuous U (1) symmetry. We consider a theory with a (non-Abelian) gauge symmetry Gg as well as the continuous U (1) symmetry, which may be gauged. This theory includes fermions with U (1) charges

15.1

Generic Aspects

187

q (f ) and representations R (f ) . Those anomalies vanish if and only if the Jacobian is trivial, i.e., J (α) = 1 for an arbitrary value of α. Using the index theorems, one finds that the anomaly-free conditions require    AU (1)−Gg −Gg ≡ (15.9) q (f ) T2 R (f ) = 0, R (f )

for the mixed U (1)–Gg –Gg anomaly, and AU (1)−grav−grav ≡



q (f ) = 0,

(15.10)

f

for the U (1)–gravity–gravity anomaly. Here, T2 (R (f ) ) is the Dynkin index of the R f representation, i.e.,        (15.11) tr ta R (f ) tb R (f ) = δab T2 R (f ) . Next, we study anomalies of the Abelian discrete symmetry, i.e., the ZN symmetry. For the ZN symmetry, we write α = 2πQN /N , where QN is the ZN charge operator and its eigenvalues are integers. Here we denote the ZN charges of the fermions (f ) by qN . Then we can evaluate the ZN –Gg –Gg and ZN –gravity–gravity anomalies in a similar way to the above U (1) anomalies. However, the important difference is that α is a discrete parameter, whereas α is a continuous parameter in the U (1) symmetry. Hence, the anomaly-free conditions, i.e., J (α) = 1, for a discrete transformation require AZN −Gg −Gg =

1  (f )   (f )  2T2 R ∈ Z, q N (f ) N

(15.12)

2  (f ) qN dim R (f ) ∈ Z, N

(15.13)

R

for the ZN –Gg –Gg anomaly, and AZN −grav−grav =

f

for the ZN –gravity–gravity anomaly. These anomaly-free conditions reduce to  (f )   (15.14a) qN T2 R (f ) = 0 mod N/2, R (f )



(f )

qN dim R (f ) = 0 mod N/2.

(15.14b)

f

Note that the Z2 symmetry is always free from the Z2 –gravity–gravity anomaly. Finally, we study anomalies of non-Abelian discrete symmetries G [3, 9]. A discrete group G comprises a finite number of elements gi . Hence, the non-Abelian discrete symmetry is anomaly-free if and only if the Jacobian vanishes for the transformation corresponding to each element gi . Furthermore, recall that (gi )Ni = 1.

188

15

Anomalies

That is, each element gi in the non-Abelian discrete group generates a ZNi symmetry. Thus, the analysis of non-Abelian discrete anomalies reduces to that of Abelian discrete anomalies. The field basis can be chosen so that gi is represented in a diago(f ) nal form. In such a basis, each field has a definite ZNi charge qNi . The anomaly-free conditions for the gi transformation are written as  (f )   qNi T2 R (f ) = 0 mod Ni /2, (15.15a) R (f )



(f )

qNi dim R (f ) = 0 mod Ni /2.

(15.15b)

f

If these conditions are satisfied for all of gi ∈ G, there are no anomalies of the full non-Abelian symmetry G. Otherwise, the non-Abelian symmetry is broken to its subgroup by quantum effects, where the subgroup does not include anomalous gi elements. Furthermore, the non-Abelian symmetry is completely broken if all elements gi ∈ G except the identity are anomalous. In principle, we can investigate anomalies of non-Abelian discrete symmetries G following the above procedure. However, we give a practically simpler way to analyze these anomalies [3, 9]. Here, we reconsider a transformation similar to (15.1) for a set of fermions Ψ = [ψ (1) , . . . , ψ (Mdα ) ], which correspond to the R M irreducible representation of the gauge group Gg and the r α irreducible representation of the non-Abelian discrete symmetry G with dimension dα . Let U correspond to one of the group elements gi ∈ G, which is represented by the matrix Dα (gi ) on r α . Then the Jacobian is proportional to its determinant det D(gi ). Thus, the representations with det Dα (gi ) = 1 do not contribute to anomalies. Therefore, non-trivial Jacobians, i.e., anomalies, originate from representations with det Dα (gi ) = 1. Note that det Dα (gi ) = det Dα (ggi g −1 ) for g ∈ G, that is, the determinant is constant on a conjugacy class. It is thus useful to calculate the determinants of elements on each irreducible representation. Such a determinant for the conjugacy class Ci can be written det(Ci )α = e

2πiq αˆ /Nˆ i Ni

,

(15.16)

on the irreducible representation r α . Note that Nˆ i is a divisor of Ni , where Ni is the order of gi in the conjugacy class Ci , i.e., g Ni = e, so the q αˆ are normalized to be Ni integers for all the irreducible representations r α . We consider the ZNˆ i symmetries and their anomalies. Then we obtain anomalyfree conditions similar to (15.15a), (15.15b). That is, the anomaly-free conditions for the conjugacy classes Ci can be written   α(f )  q ˆ T2 R (f ) = 0 mod Nˆ i /2, (15.17a) r (α) ,R (f )

 α.f

Ni

α(f ) dim R (f ) Ni

qˆ

=0

mod Nˆ i /2,

(15.17b)

15.2

Explicit Calculations

189

Table 15.1 Determinants on S3 representations

1

1

det(C1 )

1

1

1

det(C2 )

1

1

1

det(C3 )

1

−1

−1

2

for the theory including fermions with the R (f ) representations of the gauge group Gg and the r α(f ) representations of the flavor group G, which correspond to the α(f ) ZNˆ i charges q ˆ . Note that the fermion fields with the dα -dimensional represenNi

α(f ) α(f ) T2 (R (f ) ) and q ˆ dim R (f ) to these anomalies, but not Ni Ni α(f ) α(f ) dα q ˆ T2 (R (f ) ) and dα q ˆ dim R (f ) . If these conditions are satisfied for all conNi Ni

tation r α contribute q ˆ

jugacy classes of G, the full non-Abelian symmetry G is free from anomalies. Otherwise, the non-Abelian symmetry is broken by quantum effects. As we will see below, in concrete examples, the above anomaly-free conditions often lead to the same conditions between different conjugacy classes. Note that, when Nˆ i = 2, the symmetry is always free from the mixed gravitational anomalies. In what follows, we shall investigate some concrete examples of groups explicitly.

15.2 Explicit Calculations Here, we apply the above considerations of anomalies to concrete groups.

15.2.1 S3 As shown in Sect. 3.1, the group S3 has three conjugacy classes C1 = {e}, C2 = {ab, ba}, and C3 = {a, b, bab}, and three irreducible representations 1, 1 , and 2. Note that the determinants of elements are constant in a conjugacy class. The determinants of elements in singlet representations are equal to characters, and the determinants of elements in a trivial singlet representation 1 are obviously always equal to 1. On the doublet representation 2, the determinants of representation matrices in C1 , C2 , and C3 are found to be 1, 1, and −1, respectively. These determinants are shown in Table 15.1. From these results, it is found that only the conjugacy class C3 is relevant to anomalies and only the Z2 symmetry can be anomalous. Under such a Z2 symmetry, the trivial singlet has vanishing Z2 charge, while the other representations 1 and 2 have Z2 charges q2 = 1, that is, Z2 even : Z2 odd :

1, 1 , 2.

(15.18)

190 Table 15.2 Determinants on S4 representations

15

Anomalies

3

3

1

1

det(C1 )

1

1

1

1

1

det(C3 )

1

1

1

1

1

2

det(C6 )

1

−1

−1

−1

1

det(C6 )

1

−1

−1

−1

1

det(C8 )

1

1

1

1

1

The anomaly-free conditions for the Z2 –Gg –Gg mixed anomaly (15.17a), (15.17b) thus take the form       (f )  = 0 mod 1. (15.19) T2 R (f ) + T2 R 1 R (f )

2 R (f )

Note that a doublet 2 contributes to the anomaly coefficient, not by 2T2 (R (f ) ) but by T2 (R (f ) ), which is the same as 1 . To show this explicitly, we have written the summations on 1 and 2 separately.

15.2.2 S4 Similarly, we can study anomalies of S4 . As shown in Sect. 3.2, the group S4 has five conjugacy classes, C1 , C3 , C6 , C6 , and C8 , and five irreducible representations 1, 1 , 2, 3, and 3 . The determinants of group elements in each representation are shown in Table 15.2. These results imply that only the Z2 symmetry can be anomalous. Under such a Z2 symmetry, each representation has the following behavior: Z2 even : Z2 odd :

1, 3 , 1 , 2, 3.

(15.20)

The anomaly-free conditions for the Z2 –Gg –Gg mixed anomaly (15.15a), (15.15b) are thus       (f )     (f )  + = 0 mod 1. T2 R (f ) + T2 R T2 R 1 R (f )

2 R (f )

3 R (f )

(15.21)

15.2.3 A4 As shown in Sect. 4.1, there are four conjugacy classes, C1 , C3 , C4 , and C4 , and four irreducible representations 1, 1 , 1 , and 3. The determinants of group elements in

15.2

Explicit Calculations

191

Table 15.3 Determinants on A4 representations

1

1

1

3

det(C1 )

1

1

1

1

det(C3 )

1

1

1

1 1 1

det(C4 )

1

ω

ω2

det(C4 )

1

ω2

ω

Table 15.4 Determinants on A5 representations

1

3

3

4

5

det(C1 )

1

1

1

1

1

det(C15 )

1

1

1

1

1

det(C20 )

1

1

1

1

1

det(C12 )

1

1

1

1

1

 ) det(C12

1

1

1

1

1

each representation are shown in Table 15.3, where ω = e2πi/3 . These results imply that only the Z3 symmetry can be anomalous. Under such a Z3 symmetry, each representation has the following Z3 charge q3 : q3 = 0 : 1, 3, q3 = 1 : 1  , q3 = 2 : 1 .

(15.22)

This corresponds to the Z3 symmetry for the conjugacy class C4 . There is another Z3 symmetry for the conjugacy class C4 , but it is not independent of the former Z3 . The anomaly-free conditions are thus       (15.23) T2 R (f ) + 2 T2 R (f ) = 0 mod 3/2, 1 R (f )

1 R (f )

for the Z3 –Gg –Gg anomaly and  1 R (f )

dim R (f ) + 2



dim R (f ) = 0 mod 3/2,

(15.24)

1 R (f )

for the Z3 –gravity–gravity anomaly.

15.2.4 A5 We study anomalies of A5 . As shown in Sect. 4.2, there are five conjugacy classes  , and five irreducible representations 1, 3, 3 , 4, and C1 , C15 , C20 , C12 , and C12

192

15

Anomalies

5. The determinants of group elements in each representation are shown in Table 15.4. That is, the determinants of all the A5 elements are equal to unity on any representation. This result can be understood as follows. All the elements of A5 can be expressed as products of s = a and t = bab. The generators s and t are written as real matrices in all the representations 1, 3, 3 , 4, and 5. We thus find that det(t) = 1, since t 5 = e. Similarly, since s 2 = b3 = e, the possible values are det(s) = ±1 and det(b) = ωk , with k = 0, 1, 2. By imposing det(bab) = det(t) = 1, we find det(s) = det(b) = 1. Thus, it turns out that det(g) = 1 for all the A5 elements on any representation and the A5 symmetry is therefore always anomaly-free.

15.2.5 T  As shown in Chap. 5, the group T  has seven conjugacy classes C1 , C1 , C4 , C4 , C4 , C4 , and C6 , and seven irreducible representations 1, 1 , 1 , 2, 2 , 2 , and 3. The determinants of group elements on each representation are shown in Table 15.5. These results imply that only the Z3 symmetry can be anomalous. Under such a Z3 symmetry, each representation has the following Z3 charge q3 : q3 = 0 : q3 = 1 : q3 = 2 :

1, 2, 3, 1 , 2 , 1 , 2 .

(15.25)

This corresponds to the Z3 symmetry for the conjugacy class C4 . There are other Z3 symmetries for the conjugacy classes C4 , C4 , and C4 , but these are not independent of the former Z3 . The anomaly-free conditions are therefore 

        (f )  T2 R (f ) + 2 T2 R (f ) + T2 R

1 R (f )

+2





T2 R

1 R (f )

 (f )

=0

2 R (f )

mod 3/2,

(15.26)

2 R (f )

for the Z3 –Gg –Gg anomaly and 

dim R (f ) + 2

1 R (f )

+2

 2



dim R (f ) +

1 R (f )

dim R (f ) = 0 mod 3/2,

R (f )

for the Z3 –gravity–gravity anomaly.



dim R (f )

2 R (f )

(15.27)

15.2

Explicit Calculations

193

Table 15.5 Determinants on T  representations

1

1

1

2

2

2

3

det(C1 )

1

1

1

1

1

1

1

det(C1 )

1

1

1

1

1

1

1

ω

1

det(C4 )

1

ω

ω2

1

ω2

det(C4 )

1

ω2

ω

1

ω

ω2

1

1

ω

ω2

1

ω2

ω

1

det(C4 )

1

ω2

ω

1

ω

ω2

1

det(C6 )

1

1

1

1

1

1

1

det(C4 )

15.2.6 DN (N Even) We now study anomalies of DN with N even. As shown in Chap. 6, the group DN with N even has four singlets 1±± and (N/2 − 1) doublets 2k . All elements of DN can be written as products of two elements a and b. Their determinants on 2k are det(a) = 1 and det(b) = −1. Similarly, we can obtain the determinants of a and b on the four singlets 1±± . Indeed, the four singlets are classified by the values of det(b) and det(ab), that is, det(b) = 1 for 1+± , det(b) = −1 for 1−± , det(ab) = 1 for 1±+ , and det(ab) = −1 for 1±− . Thus, the determinants of b and ab are essential for anomalies. These determinants are summarized in Table 15.6. This implies that two Z2 symmetries can be anomalous. One Z2 corresponds to b and the other Z2 corresponds to ab. Under this Z2 × Z2 symmetry, each representation has the following behavior: Z2 even : 1+± , Z2 odd : 1−± , 2k ,

(15.28)

Z2 even : 1±+ , Z2 odd : 1±− , 2k .

(15.29)

The anomaly-free conditions are then 

     (f )  = 0 mod 1, T2 R (f ) + T2 R

1−± R (f )

(15.30)

2k R (f )

for the Z2 –Gg –Gg anomaly and 

     (f )  = 0 mod 1, T2 R (f ) + T2 R

1±− R (f )

for the Z2 –Gg –Gg anomaly.

2k R (f )

(15.31)

194

15

Table 15.6 Determinants on DN representations for N even

1++

1+−

Anomalies

1−+

1−−

2k

det(b)

1

1

−1

−1

−1

det(ab)

1

−1

1

−1

−1

Table 15.7 Determinants on DN representations for N odd

1+

1−

2k

det(b)

1

−1

−1

det(a)

1

1

1

15.2.7 DN (N Odd) Similarly, we study anomalies of DN with N odd. As shown in Chap. 6, the group DN with N odd has two singlets 1± and (N − 1)/2 doublets 2k . Similarly to DN with N even, all elements of DN with N odd can be written as products of two elements a and b. The determinants of a are det(a) = 1 on all representations 1± and 2k . The determinants of b are det b = 1 on 1+ and det(b) = −1 on 1− and 2k . These are shown in Table 15.7. Thus, only the Z2 symmetry corresponding to b can be anomalous. Under such a Z2 symmetry, each representation has the following behavior: Z2 even : Z2 odd :

1+ , 1− , 2 k .

The anomaly-free condition is then       (f )  = 0 mod 1, T2 R (f ) + T2 R 1− R (f )

(15.32)

(15.33)

2k R (f )

for the Z2 –Gg –Gg anomaly.

15.2.8 QN (N = 4n) We study anomalies of QN with N = 4n. As shown in Chap. 7, the group QN with N = 4n has four singlets 1±± and (N/2 − 1) doublets 2k . All elements of QN can be written as products of a and b. The determinant of a is det(a) = 1 on all the doublets 2k . On the other hand, the determinant of b is det(b) = 1 on the doublets 2k with k odd and det(b) = −1 on the doublets 2k with k even. Similarly to DN with N even, the four singlets 1±± are classified by the values of det(b) and det(ab), that is, det(b) = 1 for 1+± , det(b) = −1 for 1−± , det(b) = 1 for 1±+ , and det(b) = −1 for 1±− . Thus, the determinants of b and ab are essential for anomalies. These determinants are summarized in Table 15.8. Similarly to DN with

15.2

Explicit Calculations

195

Table 15.8 Determinants on QN representations for N/2 even

1++

1+−

1−+

1−−

2k odd

2k even

det(b)

1

1

−1

−1

1

−1

det(ab)

1

−1

1

−1

1

−1

N even, two Z2 symmetries can be anomalous. One Z2 corresponds to b and the other Z2 corresponds to ab. Under this Z2 × Z2 symmetry, each representation has the following behavior: Z2 even : Z2 odd :

1+± , 2k odd , 1−± , 2k even ,

Z2 even :

1±+ , 2k odd ,

Z2

1±− , 2k even .

odd :

(15.34) (15.35)

The anomaly-free conditions are then 

     (f )  = 0 mod 1, T2 R (f ) + T2 R

1−± R (f )

(15.36)

2k even R (f )

for the Z2 –Gg –Gg anomaly and 

     (f )  = 0 mod 1, T2 R (f ) + T2 R

1±− R (f )

(15.37)

2k even R (f )

for the Z2 –g –Gg anomaly.

15.2.9 QN (N = 4n + 2) Similarly, we study anomalies of QN with N = 4n + 2. As shown in Chap. 7, the group QN with N = 4n + 2 has four singlets 1±± and (N/2 − 1) doublets 2k . All elements of QN can be expressed as products of a and b. The determinant of a is found to be det(a) = 1 on all doublets 2k . On the other hand, the determinants of b are det(b) = 1 on the doublets 2k with k odd and det(b) = −1 on the doublets 2k with k even. For all singlets, it is found that χα (a) = χα (b2 ), i.e., det(a) = det(b2 ). This implies that the determinants of b are more important for anomalies than the determinants of a. Indeed, the determinants of b are det(b) = 1 on 1++ , det(b) = i on 1+− , det(b) = −i on 1−+ , and det(b) = −1 on 1−− . These determinants are summarized in Table 15.9. This implies that only the Z4 symmetry corresponding to b can be anomalous. Under such a Z4 symmetry, each representation has the

196

15

Table 15.9 Determinants on QN representations for N/2 odd

1++

1+−

Anomalies

1−+

1−−

2k odd

2k even

det(b)

1

i

−i

−1

1

−1

det(a)

1

−1

−1

1

1

1

following Z4 charge q4 : q4 = 0 : q4 = 1 : q4 = 2 : q4 = 3 :

1++ , 2k odd , 1+− , 1−− , 2k even , 1−+ .

(15.38)

This includes the Z2 symmetry corresponding to a and the Z2 charge q2 for each representation is defined as q2 = q4 mod 2. The anomaly-free conditions are          T2 R (f ) + 2 T2 R (f ) + 3 T2 R (f ) 1+− R (f )

+2

 



T2 R

1−− R (f )

 (f )

1−+ R (f )

= 0 mod 2,

(15.39)

2k even R (f )

for the Z4 –Gg –Gg anomaly and  1+− R (f )

+2

dim R (f ) + 2  



dim R (f ) + 3

1−− R (f )

dim R (f ) = 0 mod 2,



dim R (f )

1−+ R (f )

(15.40)

2k even R (f )

for the Z4 –gravity–gravity anomaly. Similarly, we obtain the anomaly-free condition on the Z2 symmetry corresponding to a as       (f )  = 0 mod 1, (15.41) T2 R (f ) + T2 R 1+− R (f )

1−+ R (f )

for the Z2 –Gg –Gg anomaly.

15.2.10 QD2N As shown in Chap. 8, the group QD2N has four singlets 1±± and (N/2 − 1) doublets 2k . All elements of QD2N can be written as products of a and b. The determinants of b are det(b) = −1 on all of doublets 2k . On the other hand, the determinants of a are found to be det(a) = −1 on the doublets 2k with k odd and det(a) = 1 on the doublets 2k with k even. For singlets, it is found that det(a) = 1 for 1±+ , det(a) =

15.2

Explicit Calculations

197

Table 15.10 Determinants on QD2N representations

1++

1−+

1+−

1−−

2k odd

2k even

det(b)

1

1

−1

−1

−1

−1

det(a)

1

−1

1

−1

−1

1

−1 for 1±− , det(b) = 1 for 1+± , and det(b) = −1 for 1−± . These determinants are summarized in Table 15.10. This implies that two Z2 symmetries can be anomalous. One Z2 corresponds to a and the other Z2 corresponds to b. Under this Z2 × Z2 symmetry, each representation has the following behavior: Z2 even : 1+± , 2k even , Z2 odd : 1−± , 2k odd ,

(15.42)

Z2 even : 1±+ , Z2 odd : 1±− , 2k .

(15.43)

The anomaly-free conditions are then       (f )  = 0 mod 1, T2 R (f ) + T2 R 1−± R (f )

for the Z2 –Gg –Gg anomaly and       (f )  = 0 mod 1, T2 R (f ) + T2 R 1±− R (f )

(15.44)

2k odd R (f )

(15.45)

2k R (f )

for the Z2 –Gg –Gg anomaly.

15.2.11 Σ(2N 2 ) We study anomalies of Σ(2N 2 ). As shown in Chap. 9, the group Σ(2N 2 ) has 2N singlets 1±n and N (N − 1)/2 doublets 2p,q . All elements of Σ(2N 2 ) can be written as products of a, a  , and b. Their determinants for each representation are shown in Table 15.11, where ρ = e2πi/N . We then find that only the Z2 symmetry corresponding to b and the ZN symmetry corresponding to a can be anomalous. The other ZN symmetry corresponding to a  is not independent of the ZN symmetry for a. Under the Z2 symmetry, each representation has the following behavior: Z2 even : 1+n , Z2 odd : 1−n , 2p,q ,

(15.46)

and under the ZN symmetry corresponding to a each representation has the following ZN charge qN : qN = n : qN = p + q :

1±n , 2p,q .

(15.47)

198

15

Table 15.11 Determinants on Σ(2N 2 ) representations

1+n

1−n

2k

det(b)

1

−1

−1

det(a)

ρn

ρn

ρ p+q

det(a  )

ρn

ρn

ρ p+q

The anomaly-free condition is then       (f )  = 0 mod 1, T2 R (f ) + T2 R 1−n R (f )

Anomalies

(15.48)

2p,q R (f )

for the Z2 –Gg –Gg anomaly. Similarly, the anomaly-free conditions for the ZN symmetry are       nT2 R (f ) + (p + q)T2 R (f ) = 0 mod N/2, (15.49) 1±n R (f )

2p,q R (f )

for the ZN –Gg –Gg anomaly and   n dim R (f ) + (p + q) dim R (f ) = 0 1±n R (f )

mod N/2,

(15.50)

2p,q R (f )

for the ZN –gravity–gravity anomaly.

15.2.12 Δ(3N 2 ) (N/3 = Integer) We study anomalies of Δ(3N 2 ) when N/3 = integer. As shown in Chap. 10, the group Δ(3N 2 ) with N/3 = integer has three singlets 10 , 11 , and 12 , and (N 2 − 1)/3 triplets 3[k][] . All elements of Δ(3N 2 ) can be written as products of a, a  , and b. It is found that det(a) = det(a  ) = 1 on all representations. These elements are thus irrelevant to anomalies. On the other hand, the determinant of b is found to be det(b) = 1 for all 3[k][] and 10 , det(b) = ω for 11 , and det(b) = ω2 for 12 , with ω = e2πi/3 , as shown in Table 15.12. This implies that only the Z3 symmetry corresponding to b can be anomalous. Under such a Z3 symmetry, each representation has the following Z3 charge q3 : q3 = 0 : q3 = 1 : q3 = 2 :

10 , 3[k][] , 11 , 12 .

The anomaly-free conditions are then       T2 R (f ) + 2 T2 R (f ) = 0 mod 3/2, 11 R (f )

12 R (f )

(15.51)

(15.52)

15.2

Explicit Calculations

Table 15.12 Determinants on Δ(3N 2 ) representations when N/3 = integer

199 1k

3[k][]

det(b)

ωk

1

det(a)

1

1

det(a  )

1

1

for the Z3 –Gg –Gg anomaly and  11 R (f )

dim R (f ) + 2



dim R (f ) = 0 mod 3/2,

(15.53)

12 R (f )

for the Z3 –gravity–gravity anomaly.

15.2.13 Δ(3N 2 ) (N/3 Integer) Similarly, we study anomalies of Δ(3N 2 ) when N/3 is an integer. As shown in Chap. 10, when N/3 is an integer, the group Δ(3N 2 ) has a total of nine singlets 1k, and (N 2 − 3)/3 triplets 3[k][] . All elements of Δ(3N 2 ) can be expressed as products of a, a  , and b. On all triplet representations 3[k][] , their determinants are found to be det(a) = det(a  ) = det(b) = 1. On the other hand, it is found that det(a) = det(a  ) on all nine singlets. Furthermore, the nine singlets are classified by the values of det(a) = det(a  ) and det(b). That is, the determinants of det(a) = det(a  ) and det(b) are det(a) = det(a  ) = ω and det(b) = ωk on 1k, . These results are shown in Table 15.13. This implies that two independent Z3 symmetries can be anomalous. One corresponds to b and the other corresponds to a. For the Z3 symmetry corresponding to b, each representation has the following Z3 charge q3 : q3 = 0 : q3 = 1 : q3 = 2 :

10, , 3[k][] , 11, , 12, ,

(15.54)

while for the Z3 symmetry corresponding to a, each representation has the following Z3 charge q3 : q3 = 0 : 1k,0 , 3[k][] , q3 = 1 : 1k,1 , q3 = 2 : 1k,2 .

(15.55)

The anomaly-free conditions are then  11, R (f )

     T2 R (f ) + 2 T2 R (f ) = 0 mod 3/2, 12, R (f )

(15.56)

200

15

Table 15.13 Determinants on Δ(3N 2 ) representations when N/3 is an integer

Anomalies

1k,

3[k][]

det(b)

ωk

1

det(a)

ω

1

det(a  )

ω

1

for the Z3 –Gg –Gg anomaly and   dim R (f ) + 2 dim R (f ) = 0 mod 3/2, 11, R (f )

(15.57)

12, R (f )

for the Z3 –gravity–gravity anomaly. Similarly, for the Z3 symmetry, the anomalyfree conditions are       (15.58) T2 R (f ) + 2 T2 R (f ) = 0 mod 3/2, 1k,1 R (f )

1k,2 R (f )

for the Z3 –Gg –Gg anomaly and  1k,1 R (f )

dim R (f ) + 2



dim R (f ) = 0 mod 3/2,

(15.59)

1k,2 R (f )

for the Z3 –gravity–gravity anomaly.

15.2.14 TN As shown in Chap. 11, the group TN has three singlets 10,1,2 and (N − 1)/3 triplets ¯ m . All elements of TN can be written as products of a and b, where a and b cor3(3) respond to the generators of ZN and Z3 , respectively. It is found that det(a) = 1 on all representations, so these elements are irrelevant to anomalies. On the other hand, ¯ m and det(b) = ωk the determinant of b is found to be det(b) = 1 for any triplet 3(3) for 1k (k = 0, 1, 2), as shown in Table 15.14. These results imply that only the Z3 symmetry corresponding to b can be anomalous. Under such a Z3 symmetry, each representation has the following Z3 charge q3 : q3 = 0 : q3 = 1 : q3 = 2 :

¯ 10 , 3, 3, 11 , 12 .

The anomaly-free conditions are       T2 R (f ) + 2 T2 R (f ) = 0 mod 3/2, 11 R (f )

12 R (f )

(15.60)

(15.61)

15.2

Explicit Calculations

201

Table 15.14 Determinants on TN representations

10

11

12

3m

3¯ m

det(a)

1

1

1

1

1

det(b)

1

ω

ω2

1

1

for the Z3 –Gg –Gg anomaly and 

dim R (f ) + 2



11 R (f )

dim R (f ) = 0 mod 3/2,

(15.62)

12 R (f )

for the Z3 –gravity–gravity anomaly.

15.2.15 Σ(3N 3 ) We now study anomalies of Σ(3N 3 ), which has 3N singlets 1k, and N (N 2 − 1)/3 triplets 3[][m][n] . All elements of Σ(3N 3 ) can be written as products of a, a  , a  , and b. Their determinants are shown for each representation in Table 15.15, where ρ = e2πi/N . It turns out that only the Z3 symmetry corresponding to b and the ZN symmetry corresponding to a can be anomalous. Other ZN symmetries corresponding to a  and a  are not independent of the ZN symmetry for a. For the Z3 symmetry corresponding to b, each representation has the following Z3 charge q3 : q3 = 0 : q3 = 1 : q3 = 2 :

10, , 3[][m][n] , 11, , 12, ,

(15.63)

and under the ZN symmetry corresponding to a, each representation has the following ZN charge qN : qN =  : qN =  + m + n :

1k, , 3[][m][n] .

(15.64)

The anomaly-free condition is then 

     (f )  = 0 mod 3/2, T2 R (f ) + T2 R

11, R (f )

(15.65)

12, R (f )

for the Z3 –Gg –Gg anomaly and  11, R (f )

dim R (f ) + 2

 12, R (f )

dim R (f ) = 0 mod 3/2,

(15.66)

202

15

Table 15.15 Determinants on Σ(3N 3 ) representations

Anomalies

1k,

3[][m][n]

det(b)

ωk

1

det(a)

ρ

ρ +m+n

det(a  )

ρ

ρ +m+n

det(a  )

ρ

ρ +m+n

for the Z3 –gravity–gravity anomaly. Similarly, the anomaly-free conditions for the ZN symmetry are        T2 R (f ) + ( + m + n)T2 R (f ) = 0 mod N/2, 1k, R (f )

3[][m][n] R (f )

(15.67) for the ZN –Gg –Gg anomaly and     dim R (f ) + ( + m + n) dim R (f ) = 0 mod N/2, 1k, R (f )

3[][m][n] R (f )

(15.68) for the ZN –gravity–gravity anomaly.

15.2.16 Δ(6N 2 ) (N/3 = Integer) We study anomalies of Δ(6N 2 ) when N/3 = integer. As shown in Chap. 13, when N/3 = integer, the group has two singlets 10,1 , one doublet 2, 2(N − 1) triplets 31k and 32k , and N (N − 3)/6 sextets 6[[k],[]] . All elements can be written as products of a, a  , b, and c. The determinants of a, a  , and b on any representation are found to be det(a) = det(a  ) = det(b) = 1. The determinants of c for 10 and 32k are det(c) = 1, while the other representations lead to det(c) = −1. These results are shown in Table 15.16. This implies that only the Z2 symmetry corresponding to the generator c can be anomalous. Under such a Z2 symmetry, each representation has the following Z2 charge q2 : q2 = 0 : q2 = 1 :

10 , 32k , 11 , 2, 31k , 6[[k],[]] .

(15.69)

The anomaly-free conditions are then          (f )     (f )   + + T2 R (f ) + T2 R T2 R T2 R (f ) 11 R (f )

=0

2 R (f )

mod 1,

for the Z2 –Gg –Gg anomaly.

31k R (f )

6[[k],[]] R (f )

(15.70)

15.3

Comments on Anomalies

Table 15.16 Determinants on representations of Δ(6N 2 ) with 3N = integer

203 10

11

2

31k

32k

6[[k],[]]

det(a)

1

1

1

1

1

1

det(a  )

1

1

1

1

1

1

det(b)

1

1

1

1

1

1

det(c)

1

−1

−1

−1

1

−1

Table 15.17 Determinants on representations of Δ(6N 2 ) with 3N integer

10

11

21

22

23

24

31k

32k

6[[k],[]]

det(a)

1

1

1

1

1

1

1

1

1

det(a  )

1

1

1

1

1

1

1

1

1

det(b)

1

1

1

1

1

1

1

1

1

det(c)

1

−1

−1

−1

−1

−1

−1

1

−1

15.2.17 Δ(6N 2 ) (N/3 Integer) In the same way, we study anomalies of Δ(6N 2 ) when N/3 is an integer. As shown in Chap. 13, when N/3 is an integer, the group has two singlets 10,1 , four doublets 21,2,3,4 , 2(N − 1) triplets 31k and 32k , and (N 2 − 3N + 2)/6 sextets 6[[k],[]] . All elements of Δ(54) can be written as products of a, a  , b, and c. The determinants of a, a  , and b on any representation are det(a) = det(a  ) = det(b) = 1. The determinants of c for 10 and 32k are found to be det(c) = 1, while the other representations lead to det(c) = −1. These results are shown in Table 15.17. This implies that only the Z2 symmetry corresponding to the generator c can be anomalous. Under such a Z2 symmetry, each representation has the following Z2 charge q2 : q2 = 0 : q2 = 1 :

10 , 32k , 11 , 21,2,3,4 , 31k , 6[[k],[]] .

(15.71)

The anomaly-free conditions are then          (f )     (f )   + + T2 R (f ) + T2 R T2 R T2 R (f ) 11 R (f )

=0

2k R (f )

mod 1,

31k R (f )

6[[k],[]] R (f )

(15.72)

for the Z2 –Gg –Gg anomaly. Similarly, we can analyze anomalies for other non-Abelian discrete symmetries.

15.3 Comments on Anomalies Finally, we comment on symmetry breaking by quantum effects. When a discrete (flavor) symmetry is anomalous, breaking terms can appear in the Lagrangian, e.g.,

204

15

Anomalies

by instanton effects such as 1 m Λ Φ 1 · · · Φk , Mn where Λ is a dynamical scale and M is a typical (cutoff) scale. Within the framework of string theory discrete anomalies and also anomalies of continuous gauge symmetries can be canceled by the Green–Schwarz (GS) mechanism [10], unless discrete symmetries are accidental. In the GS mechanism, dilaton and moduli fields, i.e., the so-called GS fields ΦGS , transform non-linearly under anomalous transformation. The anomaly cancellation due to the GS mechanism imposes certain relations among anomalies (see, e.g., [3]).1 Stringy non-perturbative effects, but also field-theoretical effects, induce terms in the Lagrangian such as 1 −aΦGS e Φ 1 · · · Φk . Mn The GS fields ΦGS , i.e., dilaton/moduli fields, are expected to develop non-vanishing vacuum expectation values, and the above terms correspond to breaking terms of discrete symmetries. The above breaking terms may be small. Such approximate discrete symmetries with small breaking terms may be useful in particle physics,2 if breaking terms are controllable. Alternatively, if exact symmetries are necessary, one has to arrange matter fields and their quantum numbers in such a way that models are free from anomalies.

References 1. Fujikawa, K.: Phys. Rev. Lett. 42, 1195 (1979) 2. Fujikawa, K.: Phys. Rev. D 21, 2848 (1980) 3. Araki, T., Kobayashi, T., Kubo, J., Ramos-Sanchez, S., Ratz, M., Vaudrevange, P.K.S.: Nucl. Phys. B 805, 124 (2008). arXiv:0805.0207 [hep-th] 4. Alvarez-Gaume, L., Witten, E.: Nucl. Phys. B 234, 269 (1984) 5. Alvarez-Gaume, L., Ginsparg, P.H.: Ann. Phys. 161, 423 (1985) 6. Fujikawa, K., Ojima, S., Yajima, S.: Phys. Rev. D 34, 3223 (1986) 7. Rohlin, V.: Dokl. Akad. Nauk 128, 980–983 (1959) 8. Csaki, C., Murayama, H.: Nucl. Phys. B 515, 114–162 (1998). arXiv:hep-th/9710105 9. Araki, T.: Prog. Theor. Phys. 117, 1119–1138 (2007). arXiv:hep-ph/0612306 10. Green, M.B., Schwarz, J.H.: Phys. Lett. B 149, 117–122 (1984) 11. Kobayashi, T., Nakano, H.: Nucl. Phys. B 496, 103–131 (1997). arXiv:hep-th/9612066 12. Fukuoka, H., Kubo, J., Suematsu, D.: Phys. Lett. B 678, 401 (2009). arXiv:0905.2847 [hepph]

1 See

also [11].

2 For

some applications, see, e.g., [12].

Chapter 16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

Non-Abelian discrete groups have been adopted for the flavor models of the quarks and leptons. In this chapter, we present some typical flavor models based on these discrete symmetries. The examples will illustrate how such models are built. However, before discussing the models themselves, we briefly review the main features of recent experimental data regarding neutrino flavor mixing.

16.1 Neutrino Flavor Mixing and Neutrino Mass Matrix The recent experimental data on neutrino oscillations has stimulated work on the non-Abelian discrete symmetry of flavors. Both the atmospheric neutrino mixing angle θ23 and the solar neutrino mixing angle θ12 are quite large. In particular, θ23 is almost maximal. These neutrino mixing angles are defined in the neutrino mixing matrix U by ⎛ ⎞ c13 c12 c13 s12 s13 e−iδ U = ⎝ −c23 s12 − s23 s13 c12 eiδ c23 c12 − s23 s13 s12 eiδ s23 c13 ⎠ , (16.1) iδ iδ s23 s12 − c23 s13 c12 e −s23 c12 − c23 s13 s12 e c23 c13 where cij and sij denote cos θij and sin θij , respectively. The global fit of the neutrino experimental data in Table 16.1 [1–3], indicates the tri-bimaximal mixing matrix Utribi for three lepton flavors [4–7] as follows: ⎛

√2 6 ⎜ 1 Utribi = ⎝ − √6 − √1 6

√1 3 √1 3 √1 3

0

⎞

− √1 ⎟ , 2⎠

(16.2)

√1 2

which favors the non-Abelian discrete symmetry for the lepton flavor. Indeed, various types of models leading to tri-bimaximal mixing have been proposed on the basis of non-Abelian discrete flavor symmetries, as can be seen, e.g., in [8, 9]. H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5_16, © Springer-Verlag Berlin Heidelberg 2012

205

206

16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

Table 16.1 Summary of neutrino oscillation parameters. For m231 , sin2 θ23 , and sin2 θ13 , the upper (lower) row corresponds to the normal (inverted) neutrino mass hierarchy. In [10], they assume the new reactor anti-neutrino fluxes [11] and include short-baseline reactor neutrino experiments in the fit Parameter

Best fit ±1σ

2σ

3σ

m2sol [10−5 eV2 ]

7.59+0.20 −0.18

7.24–7.99

7.09–8.19

2.50+0.09 −0.16 −(2.40+0.08 −0.09 ) +0.017 0.312−0.015

2.25–2.68 −(2.23–2.58)

2.14–2.76 −(2.13–2.67)

m2atm sin2 θ12

[10−3 eV2 ]

0.28–0.35

0.27–0.36

sin θ23

0.52+0.06 −0.07

0.41–0.61 0.42–0.61

0.39–0.64

sin2 θ13

0.013+0.007 −0.005 0.016+0.008 −0.006

0.004–0.028 0.005–0.031

0.001–0.035 0.001–0.039

2

0.52 ± 0.06

In tri-bimaximal mixing, θ13 vanishes. However, the T2K collaboration presented evidence at 2.5σ for a non-zero value of the reactor angle θ13 [12]. Quantitatively, it was found that 0.03(0.04) < sin2 2θ13 < 0.28(0.34),

90 % C.L.,

(16.3)

for |m232 | = 2.4 × 10−3 eV2 , sin2 2θ23 = 1, and δ = 0, in the normal (inverted) hierarchy of neutrino masses. Finally, Daya Bay reported the following result [13]: sin2 2θ13 = 0.092 ± 0.016 ± 0.005,

68 % C.L.

(16.4)

Thus the theoretical estimate of the neutrino mixing angles is an important subject. To begin with, we introduce typical models to reproduce the tri-bimaximal mixing of neutrino flavors in the flavor model with non-Abelian discrete symmetry. The neutrino mass matrix with tri-bimaximal mixing of flavors is expressed as the sum of three simple mass matrices in the flavor diagonal basis of the charged lepton. In terms of neutrino mass eigenvalues m1 , m2 , and m3 , the neutrino mass matrix is given by ⎛ ⎞ m1 0 0 † ∗ ⎝ 0 m2 0 ⎠ Utribi Mν = Utribi 0 0 m3 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 1 1 1 1 0 0 m1 + m3 ⎝ − m − m m m 2 1 ⎝ 1 3 ⎝ 0 1 0⎠ + 1 1 1⎠ + 0 0 1⎠. = 2 3 2 0 0 1 1 1 1 0 1 0 (16.5) This neutrino mass matrix is easily realized in some non-Abelian discrete symmetries. In the following sections, we shall thus present a simple realization which

16.2

A4 Flavor Symmetry

Table 16.2 Assignments of SU(2), A4 , and Z3 representations, where ω = e2π i/3

207 (le , lμ , lτ ) ec

μc τ c

hu,d

φl

φν

ξ

ξ  ξ 

SU(2) 2

1

1

1

2

1

1

1

1

1

A4

3

1

1

1

1

3

3

1

1

1

Z3

ω

ω2 ω2 ω2

1

1

ω

ω ω

ω

arises from the 5D non-renormalizable operators [14], or the seesaw mechanism [15–19].

16.2 A4 Flavor Symmetry Simple models realizing tri-bimaximal mixing have been proposed using the nonAbelian finite group A4 [20–67]. The A4 flavor model considered by Alterelli et al. [27, 33] realizes tri-bimaximal flavor mixing. The deviation from tri-bimaximal mixing can also be predicted. Actually, we have investigated the deviation from tribimaximal mixing including higher dimensional operators in the effective model [44, 63].

16.2.1 Realizing Tri-Bimaximal Mixing of Flavors We begin by presenting the A4 × Z3 flavor model with supersymmetry, including right-handed neutrinos [27, 33]. In the non-Abelian finite group A4 , there are twelve group elements and four irreducible representations: 1, 1 , 1 , and 3. The A4 and Z3 charge assignments of leptons, Higgs fields, and SM-singlets are listed in Table 16.2. Under the A4 symmetry, the chiral superfields for three families of the left-handed lepton doublet l = (le , lμ , lτ ) are assumed to transform according to 3, while the right-handed ones of the charged leptons ec , μc , and τ c are assigned with 1, 1 , 1 , respectively. The third row of Table 16.2 shows how each chiral multiplet transforms under Z3 , where ω = e2πi/3 . We assume flavons φl and φν which are A4 triplets. In addition to these triplet flavons, we can consider singlet flavons ξ , ξ  , ξ  , which are 1, 1 , 1 , respectively. The flavor symmetry is spontaneously broken by the vacuum expectation values (VEVs) of two 3’s, φl = (φl1 , φl2 , φl3 ), φν = (φν1 , φν2 , φν3 ), and by singlets, ξ , ξ  , ξ  , which are SU (2)L × U (1)Y singlets. In order to realize tri-bimaximal mixing, we consider the case where ξ   = ξ   = 0. The superpotential of the lepton sector which respects the gauge and the flavor symmetry is described by w = y e ec lφl hd /Λ + y μ μc lφl hd /Λ + y τ τ c lφl hd /Λ  + yφνν φν + yξν ξ + yξν ξ  + yξν ξ  llhu hu /Λ2 ,

(16.6)

where y e , y μ , y τ , yφνν , yξν , yξν , and yξν are the dimensionless coupling constants, and Λ is the cutoff scale. Hereafter, we follow the convention that the chiral superfield

208

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Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

and its lowest component are denoted by the same letter. Decompositions into the A4 singlet are given using the basis in Appendix C.2: ec lφl → ec (le φl1 + lμ φl3 + lτ φl2 ), ⎛ ⎞ 2le le − lμ lτ − lτ lμ llφν → ⎝ 2lτ lτ − le lμ − lμ le ⎠ φν 2lμ lμ − le lτ − lτ le → (2le le − lμ lτ − lτ lμ )φν1 + (2lτ lτ − le lμ − lμ le )φν3

(16.7)

+ (2lμ lμ − le lτ − lτ le )φν2 , llξ → (le le + lμ lτ + lτ lμ )ξ, llξ  → (lμ lμ + le lτ + lτ le )ξ  , llξ  → (lτ lτ + le lμ + lμ le )ξ  . We now suppose the following vacuum alignments: φl  = αl Λ(1, 0, 0),

φν  = αν Λ(1, 1, 1),

(16.8)

with ξ  = αξ Λ. We omit the discussion of the origin of these vacuum alignments, since the purpose of this section is not to present details of the model, but rather to apply the A4 group to neutrino mixing. Using these vacuum alignments and (16.7), we can obtain mass matrices for the charged leptons and neutrinos. Then, the diagonal charged lepton mass matrix is given by ⎛ e ⎞ 0 0 y Ml = α l v d ⎝ 0 y μ 0 ⎠ , (16.9) 0 0 yτ where hu,d  = vu,d . Furthermore, the effective neutrino mass matrix is given by Mν =

⎛ ⎞ 2 −1 −1 yφνξ αξ vu2 1 ⎝ −1 2 −1 ⎠ + ⎝0 Λ −1 −1 2 0 ⎞ ⎛ ⎞ ⎛ 0 0 1 1 1 1 0 1 0⎠ + b ⎝1 1 1⎠ + c ⎝0 0 0 1 1 1 1 0 1

yφνν αν vu2 3Λ ⎛

1 ⎝ =a 0 0

⎛

⎞ 0 0 0 1⎠ 1 0 ⎞ 0 1⎠, 0

(16.10)

where a=

yφνν αν vu2 Λ

,

b=−

yφνν αν vu2 3Λ

,

c=

yξν αξ vu2 Λ

.

Thus, tri-bimaximal mixing is easily derived in the A4 × Z3 flavor model.

(16.11)

16.2

A4 Flavor Symmetry

209

16.2.2 Breaking Tri-Bimaximal Mixing It should be emphasized that the A4 flavor symmetry does not necessarily give tribimaximal mixing at the leading order, even if the relevant alignments of the VEVs are realized. Certainly, for the neutrino mass matrix with three flavors, the A4 symmetry can give the mass matrix with the (2, 3) off-diagonal matrix due to the A4 singlet flavon, 1, in addition to the unit matrix and the democratic matrix, which leads to the tri-bimaximal mixing of flavors. However, the (1, 3) off-diagonal matrix and the (1, 2) off-diagonal matrix also appear at the leading order as long as the VEV of the ξ  or ξ  flavons does not vanish [45]: ⎛ ⎞ ⎛ ⎞ 0 0 1 0 1 0 ⎝ 0 1 0 ⎠ for ξ  , ⎝ 1 0 0 ⎠ for ξ  . (16.12) 1 0 0 0 0 1 Tri-bimaximal mixing is broken at the leading order in such a case. Let us consider the case of non-vanishing ξ  , but still vanishing ξ  . The charged lepton mass matrix is still diagonal, as in (16.9). The effective neutrino mass matrix is modified to ⎛ ⎞ ⎛ ⎞ 2 −1 −1 yφνξ αξ vu2 1 0 0 yφνν αν vu2 ⎝ −1 2 −1 ⎠ + ⎝0 0 1⎠ Mν = 3Λ Λ −1 −1 2 0 1 0 ⎛ ⎞ yφν  αξ  vu2 0 0 1 ξ ⎝0 1 0⎠ + Λ 1 0 0 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 1 1 1 1 0 0 0 0 1 = a ⎝0 1 0⎠ + b ⎝1 1 1⎠ + c ⎝0 0 1⎠ + d ⎝0 1 0⎠, 0 0 1 1 1 1 0 1 0 1 0 0 (16.13) where a=

yφνν αν vu2 Λ

,

b=−

yφνν αν vu2 3Λ

,

c=

yξν αξ vu2 Λ

,

d=

yξν αξ  vu2 Λ

. (16.14)

Therefore, the tri-bimaximal mixing is broken in the A4 flavor model. As can be seen from (16.13) and (16.14), the non-vanishing d is generated through the coupling llξ  hu hu . Since the relation a = −3b is given in this model, the predicted regions of the lepton mixing angles are reduced compared with the one in the previous section. In the case where the parameters a, c, d are real, they are fixed by the three neutrino masses m1 , m2 , and m3 . That is, sin θ13 can be plotted as

a function of the total mass mi .

Figure 16.1 shows the predicted sin θ13 versus mi , where the normal hierarchy of the neutrino masses is taken. The leptonic mixing is almost tri-bimaximal, that

210

16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models



Fig. 16.1 The mi dependence of sin θ13 for normal mass hierarchy, where horizontal lines denote Daya Bay data with 3 σ

Fig. 16.2 The mi dependence of sin θ13 for inverted mass hierarchy, where horizontal lines denote Daya Bay data with 3 σ

is, sin θ13 = 0, in the regime where mi  0.08–0.09 eV. In the case where m3 m2 , m1 , that is, mi  0.05 eV, sin θ

13 is expected to be around 0.15. We can also predict sin θ13 versus mi in the case of the inverted hierarchy of neutrino masses. We get a different prediction for sin θ13 , as shown in Fig. 16.2. The predicted maximal value of sin θ13 is 0.2 at mi  0.1 eV, which corresponds to m3 m2 , m1 . In conclusion, the A4 × Z3 model predicts sin θ13 = 0.15 − 0.2 for the cases with m3 m2 , m1 and m3 m2 , m1 . Finally, we comment on flavor models with other non-Abelian discrete symmetries which give the non-vanishing d effectively. One is the flavor model based on the group Δ(27), as described by Grimus and Lavoura [68]. Trimaximal mixing is enforced by the softly broken discrete symmetry. In this model, we find the relation d = eiπ/3 c, where a, b, c, d are complex. As shown in [68], a large value of sin θ13 is expected. Another example is the flavor twisting model in the 5D framework [69, 70]. In this model, flavor symmetry breaking is triggered by the boundary conditions of the bulk right-handed neutrino in the fifth spatial dimension. The parameters a, b, c, d involve the bulk neutrino masses and the volume of the extra dimension. In the case of the S4 flavor symmetry [70], there is one relation among these four parameters, so that the general allowed region is further restricted as in

16.3

S4 Flavor Model

211

the modified A4 model. By feeding in the experimental data for m2atm and m2sol , sin θ13 is predicted to be around 0.18 in the case of the normal hierarchy. For the inverted hierarchy, the predicted value of sin θ13 is almost vanishing, i.e., sin θ13 ∼ 0.

16.3 S4 Flavor Model In this section, we present a S4 × Z4 flavor model to unify the quarks and leptons in the framework of the SU(5) GUT [71]. The S4 group has 24 distinct elements and five irreducible representations 1, 1 , 2, 3, and 3 . In SU(5), matter fields are unified ¯ c , le )L dimensional representations. The 5-dimensional, into 10(q1 , uc , ec )L and 5(d ¯ 5-dimensional, and 45-dimensional Higgs of SU(5), H5 , H5¯ , and H45 are assigned 1 ¯ which are denoted by Fi , are assigned 3 of S4 . On the of S4 . Three generations of 5, other hand, the third generation of the 10-dimensional representation is assigned 1 of S4 , so that the top quark Yukawa coupling is allowed at the tree level, while the first and the second generations are assigned 2 of S4 . These 10-dimensional representations are denoted by T3 and (T1 , T2 ), respectively. These assignments of S4 for 5 and 10 lead to the completely different structure of the quark and lepton mass matrices. Right-handed neutrinos, which are SU(5) gauge singlets, are also assigned 1 and 2 for Nτc and (Nec , Nμc ), respectively. These assignments are essential to realize the tri-bimaximal mixing of neutrino flavors. Assignments of SU(5), S4 , Z4 , and U (1)FN representations are summarized in Table 16.3. Taking the vacuum alignments of the relevant gauge singlet scalars, we predict the quark mixing as well as the tri-bimaximal mixing of leptons. In particular, the Cabibbo angle is predicted to be around 15° under the relevant vacuum alignments. We introduce new scalars χi (i = 1–14), which are assumed to be SU(5) gauge singlets. Those flavons are summarized in Table 16.3. In order to obtain the natural hierarchy among quark and lepton masses, the Froggatt–Nielsen mechanism [72] is introduced as an additional U (1)FN flavor symmetry. Θ denotes the Froggatt– Nielsen flavon. The particle assignments of SU(5), S4 and Z4 , and U (1)FN are presented in Table 16.3. We can now write down the superpotential respecting the S4 , Z4 , and U (1)FN symmetries in terms of the S4 cutoff scale Λ and the U (1)FN cutoff scale Λ. The SU(5) invariant superpotential of the Yukawa sector up to the linear terms of χi is given by  wSU(5) = y1u (T1 , T2 ) ⊗ T3 ⊗ (χ1 , χ2 ) ⊗ H5 /Λ + y2u T3 ⊗ T3 ⊗ H5   + y1N Nec , Nμc ⊗ Nec , Nμc ⊗ Θ 2 /Λ¯   + y2N Nec , Nμc ⊗ Nec , Nμc ⊗ (χ3 , χ4 ) + MNτc ⊗ Nτc  ¯ + y1D Nec , Nμc ⊗ (F1 , F2 , F3 ) ⊗ (χ5 , χ6 , χ7 ) ⊗ H5 ⊗ Θ/(ΛΛ) + y2D Nτc ⊗ (F1 , F2 , F3 ) ⊗ (χ5 , χ6 , χ7 ) ⊗ H5 /Λ

212

16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

Table 16.3 Assignments of SU(5), S4 , Z4 , and U (1)FN representations (T1 , T2 )

T3

(F1 , F2 , F3 )

(Nec , Nμc )

Nτc

H5

H5¯

H45

Θ

10

10

5¯

1

1

5

5¯

45

1

S4

2

1

3

2

1

1

1

1

1

Z4

−i

−1

i

1

1

1

1

−1

1

1

0

0

1

0

0

0

0

−1

(χ1 , χ2 )

(χ3 , χ4 )

(χ11 , χ12 , χ13 )

χ14

SU(5)

U (1)FN

(χ5 , χ6 , χ7 )

(χ8 , χ9 , χ10 )

SU(5)

1

1

1

1

1

1

S4

2

2

3

3

3

1

Z4

−i

1

−i

−1

i

i

U (1)FN

−1

−2

0

0

0

−1

¯ + y1 (T1 , T2 ) ⊗ (F1 , F2 , F3 ) ⊗ (χ8 , χ9 , χ10 ) ⊗ H45 ⊗ Θ/(ΛΛ) (16.15) + y2 T3 ⊗ (F1 , F2 , F3 ) ⊗ (χ11 , χ12 , χ13 ) ⊗ H5¯ /Λ 1 , where [ ]1 denotes the only trivial singlet components of S4 extracted from tensor products. Parameters y1u , y2u , y1N , y2N , y1D , y2D , y1 , and y2 are Yukawa couplings. We take the basis II in Appendix B.2 for the multiplication rules to get the S4 singlet:  χ1 ⊗ → (T1 χ1 + T2 χ2 )1 , χ2 2 2

c

c  Ne Ne ⊗ → Nec Nec + Nμc Nμc 1 , c c Nμ 2 Nμ 2

c



c Ne Ne χ3 ⊗ ⊗ Nμc 2 Nμc 2 χ4 2

c c   Ne Nμ + Nμc Nec χ3 → ⊗ Nec Nec − Nμc Nμc χ4 2   → Nec Nμc + Nμc Nec χ3 + Nec Nec − Nμc Nμc χ4 1 , ⎛ ⎞ ⎛ ⎞

c F1 χ5 Ne ⊗ ⎝ F2 ⎠ ⊗ ⎝ χ6 ⎠ c Nμ 2 F3 3 χ7 3  1 

c √ (2F1 χ5 − F2 χ6 − F3 χ7 ) Ne 6 → ⊗ √1 (F2 χ6 − F3 χ7 ) Nμc 2

T1 T2





2

2

 → √ (2F1 χ5 − F2 χ6 − F3 χ7 ) + √ (F2 χ6 − F3 χ7 ) , 6 2 1 Nec

Nμc

16.3

S4 Flavor Model

213

⎛ ⎞ ⎞ χ5 F1 ⎝ F2 ⎠ ⊗ ⎝ χ6 ⎠ → (F1 χ5 + F2 χ6 + F3 χ7 )1 , F3 3 χ7 3 ⎞ ⎛ ⎞ ⎛

 F1 χ8 T1 ⊗ ⎝ F2 ⎠ ⊗ ⎝ χ9 ⎠ T2 2 F3 3 χ10 3  

 √1 (F2 χ9 − F3 χ10 ) T1 2 → ⊗ 1 √ (2F1 χ8 − F2 χ9 − F3 χ10 ) T2 2 6 2   T1 T2 → √ (F2 χ9 − F3 χ10 ) + √ (−2F1 χ8 + F2 χ9 + F3 χ10 ) , 6 2 1 ⎛ ⎞ ⎛ ⎞ χ8 F1 ⎝ F2 ⎠ ⊗ ⎝ χ9 ⎠ → (F1 χ8 + F2 χ9 + F3 χ10 )1 . F3 3 χ10 3 ⎛

(16.16) We discuss the quark and lepton mass matrices and flavor mixing based on this superpotential. Furthermore, we take into account the next order superpotential in the numerical study of flavor mixing and CP violation. (0) Let us start by extracting the lepton sector from the superpotential wSU(5) . Denoting Higgs doublets by hu and hd , the superpotential of the Yukawa sector respecting the S4 × Z4 × U (1)FN symmetry is given for charged leptons by  wl = −3y1

 ec μc ¯ √ (lμ χ9 − lτ χ10 ) + √ (−2le χ8 + lμ χ9 + lτ χ10 ) h45 Θ/(ΛΛ) 6 2

+ y2 τ c (le χ11 + lμ χ12 + lτ χ13 )hd /Λ.

(16.17)

For right-handed Majorana neutrinos, the superpotential is given by  wN = y1N Nec Nec + Nμc Nμc Θ 2 /Λ¯   + y2N Nec Nμc + Nμc Nec χ3 + Nec Nec − Nμc Nμc χ4 + MNτc Nτc , (16.18) and for Dirac neutrino Yukawa couplings, the superpotential is  wD =

y1D

 Nμc Nec ¯ √ (2le χ5 − lμ χ6 − lτ χ7 ) + √ (lμ χ6 − lτ χ7 ) hu Θ/(ΛΛ) 6 2

+ y2D Nτc (le χ5 + lμ χ6 + lτ χ7 )hu /Λ.

(16.19)

214

16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

Higgs doublets hu , hd and gauge singlet scalars Θ and χi are assumed to develop their VEVs as follows: hd  = vd , h45  = v45 , Θ = θ, hu  = vu ,     (χ5 , χ6 , χ7 ) = (u5 , u6 , u7 ), (χ3 , χ4 ) = (u3 , u4 ),     (χ8 , χ9 , χ10 ) = (u8 , u9 , u10 ), (χ11 , χ12 , χ13 ) = (u11 , u12 , u13 ),

(16.20)

which are assumed to be real. We then obtain the mass matrix for charged leptons as √ √ ⎞ ⎛ ⎛ ⎞ 0 0 0 0 √ α9 /√2 −α10 /√ 2 0 0 ⎠, Ml = −3y1 λv45 ⎝ −2α8 / 6 α9 / 6 α10 / 6 ⎠ + y2 vd ⎝ 0 α11 α12 α13 0 0 0 (16.21) while the right-handed Majorana neutrino mass matrix is given by ⎛

y1N λ2 Λ¯ + y2N α4 Λ ⎝ MN = y2N α3 Λ 0

⎞ y2N α3 Λ 0 y1N λ2 Λ¯ − y2N α4 Λ 0 ⎠ . 0 M

(16.22)

Note that the (1, 3), (2, 3), (3, 1), and (3, 3) elements of the right-handed Majorana neutrino mass matrix vanish. These are the so-called SUSY zeros. The Dirac mass matrix for the neutrinos is √ √ √ ⎞ ⎛ ⎛ ⎞ 2α5 / 6 −α6 /√ 6 −α7 /√6 0 0 0 0 0 ⎠, MD = y1D λvu ⎝ 0 α6 / 2 −α7 / 2 ⎠ + y2D vu ⎝ 0 α α α 0 0 0 5 6 7 (16.23) ¯ where we denote αi ≡ ui /Λ and λ ≡ θ/Λ. If we can take the vacuum alignment to be (u8 , u9 , u10 ) = (0, u9 , 0),

(u11 , u12 , u13 ) = (0, 0, u13 ),

that is, α8 = α10 = α11 = α12 = 0, we obtain ⎛

0 Ml = ⎝ 0 0

√ ⎞ 0 −3y1 λα9 v45 /√2 ⎠, −3y1 λα9 v45 / 6 0 0 y2 α13 vd

(16.24)

and Ml† Ml is then obtained as follows: ⎛

0 0 Ml† Ml = vd2 ⎝ 0 6|y¯1 λα9 |2 0 0

⎞ 0 ⎠, 0 2 |y2 |2 α13

(16.25)

16.3

S4 Flavor Model

215

where we replace y1 v45 by y¯1 vd . That is, the left-handed mixing angles of the charged lepton mass matrix vanish. The charged lepton masses are given by m2e = 0,

m2μ = 6|y¯1 λα9 |2 vd2 ,

2 2 m2τ = |y2 |2 α13 vd .

(16.26)

It is remarkable that the electron mass vanishes. The electron mass is obtained in the next order. Taking the vacuum alignment (u3 , u4 ) = (0, u4 ) and (u5 , u6 , u7 ) = (u5 , u5 , u5 ) in (16.22), the right-handed Majorana mass matrix for the neutrinos turns out to be ⎛ N 2 ⎞ y1 λ Λ¯ + y2N α4 Λ 0 0 MN = ⎝ (16.27) 0 y1N λ2 Λ¯ − y2N α4 Λ 0 ⎠ , 0 0 M and the Dirac mass matrix for the neutrinos is √ √ √ ⎞ ⎛ ⎛ 2α5 / 6 −α5 /√ 6 −α5 /√6 0 MD = y1D λvu ⎝ 0 α5 / 2 −α5 / 2 ⎠ + y2D vu ⎝ 0 α5 0 0 0

⎞ 0 0 ⎠. α5 (16.28) T M −1 M , the left-handed Majorana neuUsing the seesaw mechanism Mν = MD D N trino mass matrix can be written as ⎞ ⎛ a − 13 b a − 13 b a + 23 b ⎟ ⎜ 1 1 1 1 1 ⎟ (16.29) Mν = ⎜ ⎝a − 3b a + 6b + 2c a + 6b − 2c⎠, a − 13 b a + 16 b − 12 c a + 16 b + 12 c 0 0 α5

where a=

(y2D α5 vu )2 , M

b=

(y1D α5 vu λ)2 , y N λ2 Λ¯ + y N α4 Λ 1

c=

2

(y1D α5 vu λ)2 . y1N λ2 Λ¯ − y2N α4 Λ (16.30)

The neutrino mass matrix is decomposed as ⎛ ⎞ ⎛ ⎞ ⎛ 1 0 0 1 1 1 1 b+c ⎝ 3a − b ⎝ b−c ⎝ 0 1 0⎠+ 1 1 1⎠+ 0 Mν = 2 3 2 0 0 1 1 1 1 0

0 0 1

⎞ 0 1 ⎠ , (16.31) 0

which gives the tri-bimaximal mixing matrix Utribi and mass eigenvalues as follows: m1 = b,

m2 = 3a,

m3 = c.

(16.32)

We now discuss the quark sector. For down-type quarks, we can write the superpotential as follows:   1  1  c c c c c ¯ wd = y1 √ s χ9 − b χ10 q1 + √ −2d χ8 + s χ9 + b χ10 q2 h45 Θ/(ΛΛ) 6 2  + y2 d c χ11 + s c χ12 + bc χ13 q3 hd /Λ. (16.33)

216

16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

Since the vacuum alignment is fixed in the lepton sector, as can be seen from (16.20), the down-type quark mass matrix is given to leading order by ⎛ ⎞ 0 √ 0 √ 0 (16.34) Md = vd ⎝ y¯1 λα9 / 2 y¯1 λα9 / 6 0 ⎠, 0 0 y2 α13 where we denote y¯1 vd = y1 v45 . Then, we have ⎛ 1 1 2 √ |y¯ λα9 |2 2 |y¯1 λα9 | 2 3 1 † 2⎜ 1 1 2 Md Md = vd ⎝ √ |y¯1 λα9 |2 6 |y¯1 λα9 | 2 3 0 0

0

⎞

⎟ ⎠. 0 2 2 |y2 | α13

This matrix can be diagonalized by the orthogonal matrix ⎛ ⎞ cos 60° sin 60° 0 Ud = ⎝ − sin 60° cos 60° 0 ⎠ . 0 0 1

(16.35)

(16.36)

The down-type quark masses are given by m2d = 0,

2 m2s = |y¯1 λα9 |2 vd2 , 3

2 2 m2b ≈ |y2 |2 α13 vd ,

(16.37)

which correspond to the charged lepton masses in (16.26). The down quark mass vanishes, like the electron mass, but tiny masses appear in the next order. A realistic CKM mixing matrix is obtained in the quark sector by including the next order terms of the superpotential, in which terms quadratic in the χi , such as d c q1 χ1 χ5 , are dominant. We obtain the next order down-type quark mass matrix elements ¯ij , which are given in terms of Yukawa couplings and VEVs of flavons. The magnitudes of the ¯ij ’s are O(α˜ 2 ), where α˜ is a linear combination of αi ’s. The down-type quark mass matrix is written in terms of ¯ij as ⎛ ⎞ ¯21 ¯31 ¯11 √ Md  ⎝ 3ms + ¯12 ms + ¯22 (16.38) ¯32 ⎠ , 2 2 ¯13 ¯23 mb + ¯33 where ij should be the order of md . By rotating Md† Md with the mixing matrix Ud in (16.36), we have ⎛ ⎞ O(m2d ) O(md ms ) O(md mb ) Ud† Md† Md Ud  ⎝ O(md ms ) O(m2s ) O(md mb ) ⎠ . m2b O(md mb ) O(md mb )

(16.39)

d , θ d , θ d in the mass matrix of (16.39) as Then we get the mixing angles θ12 13 23   

md md md d d d θ12 = O , θ13 = O , θ23 = O , (16.40) ms mb mb

16.3

S4 Flavor Model

217

where CP violating phases are neglected. We now consider the up-type quark sector. Here the superpotential respecting S4 × Z4 × U (1)FN is given by  wu = y1u uc χ1 + cc χ2 q3 + t c (q1 χ1 + q2 χ2 ) hu /Λ + y2u t c q3 hu .

(16.41)

We denote their VEVs by 

 (χ1 , χ2 ) = (u1 , u2 ).

(16.42)

We then obtain the mass matrix for up-type quarks as ⎛

0 Mu = vu ⎝ 0 y1u α1

0 0 y1u α2

⎞ y1u α1 y1u α2 ⎠ . y2u

(16.43)

The next order terms of the superpotential are also important to predict the CP violation in the quark sector. The relevant superpotential is given to the next highest order by  u wu = y (T1 , T2 ) ⊗ (T1 , T2 ) ⊗ (χ1 , χ2 ) ⊗ (χ1 , χ2 ) ⊗ H5 /Λ2 a u + y (T1 , T2 ) ⊗ (T1 , T2 ) ⊗ χ14 ⊗ χ14 ⊗ H5 /Λ2 b

u + y T ⊗ T3 ⊗ (χ8 , χ9 , χ10 ) ⊗ (χ8 , χ9 , χ10 ) ⊗ H5 /Λ2 1 . (16.44) c 3 The multiplication rule to get the S4 singlet is

T1 T2



T1 ⊗ T2 2



χ1 ⊗ χ2 2



χ1 ⊗ χ 2 2

 2

→ (T1 T1 + T2 T2 )1 ⊗ (χ1 χ1 + χ2 χ2 )1  

T1 T2 + T2 T1 χ1 χ2 + χ2 χ1 ⊕ ⊗ T1 T1 − T2 T2 2 χ1 χ1 − χ2 χ2 2  → (T1 T1 + T2 T2 )(χ1 χ1 + χ2 χ2 ) 1  ⊕ (T1 T2 + T2 T1 )(χ1 χ2 + χ2 χ1 ) + (T1 T1 − T2 T2 )(χ1 χ1 − χ2 χ2 ) 1 .

(16.45)

We obtain the following mass matrix including the next order terms: ⎛

2y u α 2 + y u α 2 ⎜ a1 u1 2b 14 ya2 α1 Mu = vu ⎝ y1u α1

u α2 y a2 1 u u α2 2ya1 α12 + y b 14 y1u α1

⎞ y1u α1 ⎟ y1u α1 ⎠, u u 2 y2 + yc α9

(16.46)

218

16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

where we take the alignment α1 = α2 . After rotating the mass matrix Mu through θ12 = 45°, we get ⎛ ⎞ u − y u )α 2 + y u α 2 (2y 0 0   1 14 b a1 a2 √ u ⎟ ⎜ u + y u )α 2 + y u α 2 Mˆ u ≈ vu ⎝ 0 (2y 2y1 α1 ⎠ . b 14 a1 √a2 u 1 0 2y1 α1 y2u (16.47) This mass matrix is taken to be real by removing phases. The matrix is diagonalized by the orthogonal transformation VuT Mˆ u VF , where ⎞ ⎛   1 0 0 mc mt ⎠ ⎝ rc , rc = Vu  0 rt , rt = . (16.48) mc + mt mc + mt 0 −r r c

t

Now we can calculate the CKM matrix. Mixing matrices of up- and down-type quarks are summarized by ⎞ ⎛ ⎞⎛ ⎞⎛ 1 0 0 cos 45° sin 45° 0 1 0 0 rc ⎠ , Uu  ⎝ − sin 45° cos 45° 0 ⎠ ⎝ 0 e−iρ 0 ⎠ ⎝ 0 rt 0 −rc rt 0 0 1 0 0 1 ⎛ ⎞⎛ ⎞ d d 1 θ12 θ13 cos 60° sin 60° 0 d − θd θd d ⎠. Ud  ⎝ − sin 60° cos 60° 0 ⎠ ⎝ −θ12 1 θ23 13 23 d d d d d d 0 0 1 −θ13 + θ12 θ23 −θ23 − θ12 θ13 1 (16.49) Therefore, the CKM matrix is given by Uu† Ud . The relevant mixing elements are d cos 15° + sin 15°, Vus ≈ θ12 d d Vub ≈ θ13 cos 15 ° + θ23 sin 15°, d iρ d iρ Vcb ≈ −rt θ13 e sin 15° + rt θ23 e cos 15 ° − rc ,   d d d d d d Vtd ≈ −rc sin 15°eiρ − rc θ12 + θ13 θ23 eiρ cos 15° + rt −θ13 + θ12 θ23 .

(16.50)

We can reproduce the experimental values with a parameter set ρ = 123°,

d θ12 = −0.0340,

d θ13 = 0.00626,

d θ23 = −0.00880, (16.51)

putting typical GUT scale masses mu = 1.04 × 10−3 GeV, mc = 302 × 10−3 GeV, mt = 129 GeV [73]. In terms of the phase ρ, we can also estimate the magnitude of CP violation through the Jarlskog invariant JCP [74], which is given by    ∗  ≈ 3.06 × 10−5 . |JCP | = Im Vus Vcs∗ Vub Vcb (16.52) Our prediction is consistent with experimental values JCP = 3.05+0.19 −0.20 .

16.4

Alternative Flavor Mixing

Table 16.4 Assignments of SU(2) and A5 representations

219 L

e¯

H

ξ

ψ

χ

SU(2)

2

1

2

1

1

1

A5

3

3

1

5

5

4

16.4 Alternative Flavor Mixing In the previous section, we presented flavor models reproducing tri-bimaximal mixing of lepton flavors. However, there are other flavor mixing patterns like the golden ratio, trimaximal, and bimaximal for lepton flavor mixing. Let us begin with the golden ratio which appears in the solar mixing angle θ12 . and D10 [76] models. One example The golden ratio can be derived from the A5 [75]√ is proposed as tan θ12 = 1/φ, where φ = (1 + 5)/2  1.62 [77]. The rotational icosahedral group, which is isomorphic to A5 , the alternating group of five elements, provides a natural context for the golden ratio cos θ12 = φ/2 [75]. In this model, the solar angle is related to the golden ratio, the atmospheric angle is maximal, and the reactor angle vanishes. The particle contents are summarized in Table 16.4. In this model, ξ couples to the LH LH operator and ψ and χ couple to the LeH ¯ operator. This setup is derived from additional symmetries such as Zn , which forbid ξ and ψ from coupling to LeH ¯ and LH LH , respectively. For simplicity, it is assumed that the tree level LL term is forbidden by such additional symmetries. With these assumptions, the mass terms can be written as follows: Lmass =

αij k βij k γij l Li H Lj H ξk +  Li e¯j H ψk +  Li e¯j H χl + h.c., MM  M M

(16.53)

in which M  represents the scale of flavor symmetry breaking, and αij k , βij k , and γij l are dimensionless couplings that encode the tensor product decomposition of the icosahedral symmetry. In principle, M  bears no relation to M. Taking the VEV alignment to be ξ  =

√  2 3 1 − m ), − m ), 0, 0, −(m + m ) (m (m √ 2 1 √ 2 1 1 2 , 2α 15 15

(16.54)

the neutrino mass matrix becomes ⎛

φm1 + φ1 m2 1 ⎜ Mν = √ ⎝ m2 − m1 5 0

m2 − m1 1 φ m1 + φm2 0

⎞ 0 ⎟ 0 ⎠, √ − 5(m1 + m2 )

(16.55)

√ where φ = (1 + 5)/2. The neutrino mass eigenvalues are then m1 , m2 , and m3 = −(m1 + m2 ). The neutrino mixing matrix Uν , defined by Uν Mν UνT , satisfies

220

16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

θ12 = tan−1 (1/φ) = 31.72°:

⎛

√φ ⎜ 5 Uν = ⎜ ⎝ √15φ

 − √1  5φ √φ 5

0

0

0

⎞

⎟ ⎟ 0 ⎠. −i

(16.56)

The maximal mixing between the second and third families is derived from the charged lepton sector. To leading order in the flavon fields, only mτ is nonvanishing. Taking the VEV alignment to be

   1 mτ 5 2 mτ 0, , 0, 1 , (16.57) − χ = √ , 0, − √ φ, 0, 1 , ψ = √ 3 φ 2 6β 3 3 2γ the charged lepton mass matrix takes the form ⎛ ⎞ 0 0 0 1 ⎝ 0 0 mτ ⎠ . Me = √ 2 0 0 mτ

(16.58)

The left-handed states are diagonalized by the mixing matrix Ue given by ⎞ ⎛ 1 0 0 1 1 ⎟ ⎜ Ue = ⎝ 0 √2 − √2 ⎠ . (16.59) √1 0 √1 2

2

In (16.56) and (16.59), the lepton mixing matrix U is obtained as  ⎛  ⎞ √φ √1 0 ⎜ ⎟ 5φ 5 ⎜ ⎟ U = Ue Uν† = ⎜ − √1 √15φ √1 √φ5 − √1 ⎟ P, 2 2 2⎠ ⎝ √1 √φ √1 − √1 √1 2

5φ

2

5

(16.60)

2

where P = Diag(1, 1, i) is the Majorana phase matrix. In conclusion, the lepton mixing matrix has a vanishing reactor mixing angle, a maximal atmospheric mixing angle, and a solar angle given by θ12 = tan−1 (1/φ). We now discuss another mixing pattern, namely, trimaximal lepton mixing [68], defined by |Uα2 |2 = 1/3 for α = e, μ, whence the mixing matrix Utri is given using an arbitrary angle θ and a phase η by ⎞⎛ ⎛ 2 ⎞ √ √1 0 cos θ 0 sin θ e−iη 6 3 ⎜ 1 1 ⎟ 1 ⎠. 0 1 0 Utri = ⎝ − √6 √3 − √2 ⎠ ⎝ (16.61) iη 0 1 1 1 − sin θ e cos θ √ √ −√ 6

3

2

This corresponds to a two-parameter lepton flavor mixing matrix. In Sect. 16.2.2, we presented a model for the lepton sector in which trimaximal mixing is based on

16.4

Alternative Flavor Mixing

221

Table 16.5 Assignments of SU(2), S4 , Z4 , and U (1)FN representations l ec μc τc hu,d θ ϕl

χl

ξν

ϕν

SU(2)

2

1

1

1

2

1

1

1

1

1

S4

3

1

1

1

1

1

3

3

1

3

Z4

1

−1

−i

−i

1

1

i

i

1

1

U (1)FN

0

2

1

0

0

−1

0

0

0

0

the group A4 , or the group (27). An interesting feature of the (27) model is that no vacuum alignment is required. The other model is based on the S3 [69] or S4 [70] discrete symmetry, where the symmetry breaking is triggered by the boundary conditions of the bulk right-handed neutrino in the fifth spatial dimension. Finally, we consider bimaximal mixing [78], with solar angle θ12 = π/4, atmospheric angle θ23 = π/4, and reactor angle θ13 = 0. The mixing matrix Ubi is ⎞ ⎛ 1 √ √1 0 2 2 ⎜ 1 1 − √1 ⎟ Ubi = ⎝ − 2 . (16.62) 2 2⎠ 1 √1 − 12 2 2

The S4 × Z4 flavor model is presented in [79]. The particle contents are summarized in Table 16.5. In the charged lepton sector, the superpotential is given by (1)

wl =

(2)

(3)

ye θ 2 c ye θ 2 c ye θ 2 c e (lϕ ϕ )h + e (lχ χ )h + e (lϕl χl )hd l l d l l d Λ2 Λ2 Λ2 Λ2 Λ2 Λ2 yμ θ c yτ μ (lχl ) hd + τ c (lϕl )hd , + (16.63) ΛΛ Λ

(i)

where ye , yμ , and yτ are Yukawa couplings. In the neutrino sector, the effective superpotential is wνeff =

M a b (lhu lhu ) + 2 (lhu lhu )ξν + 2 (lhu lhu ϕl ), Λ Λ Λ

(16.64)

where M, a, and b are given in mass units. Taking the VEV alignment and VEVs to be ϕl  χl  = A(0, 1, 0), = B(0, 0, 1), Λ Λ θ  ξν  = D, = t, hu,d  = vu,d , Λ Λ the charged lepton mass matrix is diagonal: ⎛ (1) (2) (3) (ye B 2 − ye A2 + ye AB)t 2 ⎝ ml = 0 0

ϕν  = C(0, 1, −1), Λ

0 yμ Bt 0

⎞ 0 0 ⎠ vd , yτ A

(16.65)

(16.66)

222

16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

and the effective neutrino mass matrix is ⎛ 2M + 2aD −2bC ⎝ −2bC 0 = meff ν −2bC 2M + 2aD

⎞ −2bC v2 2M + 2aD ⎠ u . Λ 0

(16.67)

The neutrino mixing is then bimaximal, as can be seen from (16.62), and the neutrino mass eigenvalues are m1 = 2|M + aD −

√

2bC|

vu2 , Λ

m2 = 2|M + aD +

v2 m3 = 2|M + aD| u . Λ

√ v2 2bC| u , Λ

(16.68)

Note that bimaximal neutrino mixing is also studied in the context of the quark– lepton complementarity of mixing angles [80] in the S4 model [81].

16.5 Comments on Other Applications Supersymmetric extension is an interesting candidate for physics beyond the standard model. Even if the theory is supersymmetric at high energy, supersymmetry must break above the weak scale. Supersymmetry breaking induces soft supersymmetry breaking terms such as gaugino masses, sfermion masses, and scalar trilinear couplings, i.e., the so-called A-terms. Flavor symmetries control not only quark/lepton mass matrices, but also squark/slepton masses and their A-terms. Suppose the flavor symmetries are exact. When the three families have different quantum numbers under flavor symmetries, the squark/slepton mass-squared matrices are diagonal. Furthermore, when three (two) of the three families correspond to triplets (doublets) of flavor symmetries, their diagonal squark/slepton masses are degenerate. That is, the sfermion mass-squared matrix (m2 )ij is ⎛ 2 m  2 m ij = ⎝ 0 0

0 m2 0

⎞ 0 0 ⎠, m2

(16.69)

when the three families correspond to a triplet. On the other hand, the sfermion mass-squared matrix (m2 )ij becomes ⎛ 2 m  2 m ij = ⎝ 0 0

0 m2 0

⎞ 0 0 ⎠, m2

(16.70)

when the first two families correspond to a doublet and the third family corresponds to a singlet. These patterns would become an interesting prediction of a certain

16.6

Comment on Origins of Flavor Symmetries

223

class of flavor models, and it could be tested if the supersymmetry breaking scale is reachable by collider experiments. Flavor symmetries have similar effects on A-terms. These results are very important to suppress flavor changing neutral currents, which are strongly constrained by experiments. However, the flavor symmetry must break to lead to realistic quark/lepton mass matrices. Such breaking effects deform the above predictions. How much results are changed depends on breaking patterns. If masses of superpartners are O(100) GeV, some models may be ruled out, e.g., by experiments on flavor changing neutral currents (see, e.g., [24, 52, 82–85]). The application to dark matter (DM) models is another interesting topic. It is known that DM should have a global symmetry to be stabilized or long-lived, even after electroweak breaking. This kind of symmetry could arise naturally from the non-Abelian flavor symmetries [86–92]. Indirect detections of DM were recently reported by PAMELA [93] and FermiLAT [94], where the positron excess and the total flux of electrons and positrons are observed in cosmic rays. These observations can be explained by scattering/decay of TeV-scale DM particles. Since PAMELA measured negative results for the antiproton excess [95], leptophilic DM is preferable. Even so, if the main final state of scattering or decay of DM is τ ± , this annihilation/decay mode is disfavored because it will overproduce gamma-rays in the final radiation state [96]. This may indicate that, if the cosmic-ray anomalies are induced by DM scattering or decay, these processes also reflect the flavor structure of the model. In decaying DM models, no anti-proton excess in cosmic rays implies that the lifetime of the DM particle should be O(1026 ) s. This long lifetime is achieved if the TeV-scale DM (e.g., a gauge singlet fermion X) decays into leptons by dimension 2 suppressed by GUT scale Λ ∼ 1016 GeV. In this case, ¯ LX/Λ ¯ six operators LE the lifetime of the DM is estimated as Γ −1 ∼ [(TeV)5 /Λ4 ]−1 ∼ 1026 s. However, it could be difficult in general to induce only such an operator and forbid the other ¯ undesirable operators, e.g., LEX or other four-dimensional interacting terms of X. On the other hand, the non-Abelian discrete symmetries can play an important role in well selecting the operator [97, 98]. [Moreover, it has been shown in [97] that it is impossible to adapt U (1) symmetries.] The bosonic DM case can be considered in a similar way [99–101].

16.6 Comment on Origins of Flavor Symmetries What is the origin of non-Abelian flavor symmetries? Some of them are symmetries of geometrical solids, so their origin may be geometrical symmetries in extra dimensions. For example, it is found that the two-dimensional orbifold T 2 /Z2 with proper values of moduli has discrete symmetries such as A4 and S4 [102, 103] (see also [104]). Superstring theory is a promising candidate for a unified theory including gravity, and predicts an extra six dimensions. Superstring theory on a certain type of

224

16

Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models

six-dimensional compact space realizes a discrete flavor symmetry. Such a string theory leads to stringy selection rules for allowed couplings among matter fields in four-dimensional effective field theory. Such stringy selection rules and geometrical symmetries along with broken (continuous) gauge symmetries result in discrete flavor symmetries in superstring theory. For example, discrete flavor symmetries in heterotic orbifold models are studied in [83, 105, 106], and D4 and Δ(54) are realized. Magnetized/intersecting D-brane models also realize the same flavor symmetries, together with other types such as Δ(27) [107–109]. Different types of nonAbelian flavor symmetries may be derived in other string models. These studies are thus quite important. Alternatively, discrete flavor symmetries may originate from continuous (gauge) symmetries [110–112]. At any rate, the experimental data of quark/lepton masses and mixing angles have no symmetry. Thus, non-Abelian flavor symmetries must be broken to reproduce the experimentally observed masses and mixing angles. The breaking direction is important, because the forms of mass matrices are determined by the direction along which the flavor symmetries break. Hence, we need a proper breaking direction to derive realistic values of quark/lepton masses and mixing angles. One way to fix the breaking direction is to analyze the potential minima of scalar fields with non-trivial representations of flavor symmetries. The number of potential minima may be finite, and realistic breaking would occur in one of them. At least, this is the conventional approach. Another scenario to fix the breaking direction could be realized in theories with extra dimensions. One can impose the boundary conditions of matter fermions [69] and/or flavon scalars [113–115] in bulk in such a way that zero modes for some components of the irreducible multiplets are projected out, that is, breaking the symmetry. If a proper component of a flavon multiplet remains, that can realize a realistic breaking direction.

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86. Hirsch, M., Morisi, S., Peinado, E., Valle, J.W.F.: Phys. Rev. D 82, 116003 (2010). arXiv:1007.0871 [hep-ph] 87. Meloni, D., Morisi, S., Peinado, E.: Phys. Lett. B 697, 339–342 (2011). arXiv:1011.1371 [hep-ph] 88. Boucenna, M.S., Hirsch, M., Morisi, S., Peinado, E., Taoso, M., Valle, J.W.F.: J. High Energy Phys. 1105, 037 (2011). arXiv:1101.2874 [hep-ph] 89. Morisi, S.: J. Phys. Conf. Ser. 259, 012094 (2010) 90. Meloni, D., Morisi, S., Peinado, E.: Phys. Lett. B 703, 281–287 (2011). arXiv:1104.0178 [hep-ph] 91. Toorop, R.d.A., Bazzocchi, F., Morisi, S.: arXiv:1104.5676 [hep-ph] 92. Batell, B., Pradler, J., Spannowsky, M.: J. High Energy Phys. 1108, 038 (2011). arXiv:1105.1781 [hep-ph] 93. Adriani, O., et al. (PAMELA Collaboration): Nature 458, 607 (2009). arXiv:0810.4995 [astro-ph] 94. Abdo, A.A., et al. (Fermi LAT Collaboration): Phys. Rev. Lett. 102, 181101 (2009). arXiv:0905.0025 [astro-ph.HE] 95. Adriani, O., Barbarino, G.C., Bazilevskaya, G.A., Bellotti, R., Boezio, M., Bogomolov, E.A., Bonechi, L., Bongi, M., et al.: Phys. Rev. Lett. 102, 051101 (2009). arXiv:0810.4994 [astroph] 96. Papucci, M., Strumia, A.: J. Cosmol. Astropart. Phys. 1003, 014 (2010). arXiv:0912.0742 [hep-ph] 97. Haba, N., Kajiyama, Y., Matsumoto, S., Okada, H., Yoshioka, K.: Phys. Lett. B 695, 476–481 (2011). arXiv:1008.4777 [hep-ph] 98. Kajiyama, Y., Okada, H.: Nucl. Phys. B 848, 303–313 (2011). arXiv:1011.5753 [hep-ph] 99. Daikoku, Y., Okada, H., Toma, T.: arXiv:1106.4717 [hep-ph] 100. Kajiyama, Y., Okada, H., Toma, T.: arXiv:1109.2722 [hep-ph] 101. Kajiyama, Y., Okada, H., Toma, T.: Eur. Phys. J. C 71, 1688 (2011). arXiv:1104.0367 [hepph] 102. Altarelli, G., Feruglio, F., Lin, Y.: Nucl. Phys. B 775, 31 (2007). arXiv:hep-ph/0610165 103. Adulpravitchai, A., Blum, A., Lindner, M.: J. High Energy Phys. 0907, 053 (2009). arXiv:0906.0468 [hep-ph] 104. Abe, H., Choi, K.-S., Kobayashi, T., Ohki, H., Sakai, M.: Int. J. Mod. Phys. A 26, 4067–4082 (2011). arXiv:1009.5284 [hep-th] 105. Kobayashi, T., Raby, S., Zhang, R.J.: Nucl. Phys. B 704, 3 (2005). arXiv:hep-ph/0409098 106. Kobayashi, T., Nilles, H.P., Ploger, F., Raby, S., Ratz, M.: Nucl. Phys. B 768, 135 (2007). arXiv:hep-ph/0611020 107. Abe, H., Choi, K.S., Kobayashi, T., Ohki, H.: Nucl. Phys. B 820, 317 (2009). arXiv:0904.2631 [hep-ph] 108. Abe, H., Choi, K.S., Kobayashi, T., Ohki, H.: Phys. Rev. D 80, 126006 (2009). arXiv:0907.5274 [hep-th] 109. Abe, H., Choi, K.S., Kobayashi, T., Ohki, H.: arXiv:1001.1788 [hep-th] 110. de Medeiros Varzielas, I., King, S.F., Ross, G.G.: Phys. Lett. B 644, 153 (2007). arXiv:hep-ph/0512313 111. Adulpravitchai, A., Blum, A., Lindner, M.: J. High Energy Phys. 0909, 018 (2009). arXiv:0907.2332 [hep-ph] 112. Frampton, P.H., Kephart, T.W., Rohm, R.M.: Phys. Lett. B 679, 478 (2009). arXiv:0904.0420 [hep-ph] 113. Kobayashi, T., Omura, Y., Yoshioka, K.: Phys. Rev. D 78, 115006 (2008). arXiv:0809.3064 114. Seidl, G.: Phys. Rev. D 81, 025004 (2010). arXiv:0811.3775 [hep-ph] 115. Adulpravitchai, A., Schmidt, M.A.: arXiv:1001.3172 [hep-ph]

Appendix A

Useful Theorems

In this appendix, we give simple proofs of useful theorems (see also, e.g., [1–4]). Lagrange’s Theorem The order NH of a subgroup of a finite group G is a divisor of the order NG of G. Proof If H = G, the claim is trivial since NH = NG . Thus, we consider H = G. Let a1 be an element of G, but not contained in H . Here, we denote all elements in H by {e = h0 , h1 , . . . , hNH −1 }. Then we consider the products of a1 and elements of H : a1 H = {a1 , a1 h1 , . . . , a1 hNh −1 }.

(A.1)

All of a1 hi are different from each other. None of a1 hi are contained in H , since if a1 hi = hj , we would find a1 = hj h−1 i , that is, a1 would be an element in H after all. Thus, the set a1 H contains NH elements. Next let a2 be an element of G, but contained in neither H nor a1 H . If a2 hi = a1 hj , the element a2 could be written as a2 = a1 hj h−1 / H and i , that is, it would be an element of a1 H . Thus, when a2 ∈ a2 ∈ / a1 H , the set a2 H yields NH new elements. We repeat this process. Then we can decompose G = H + a1 H + · · · + am−1 H. This implies NG = mNH .

(A.2) 

Theorem For a finite group, every representation is equivalent to a unitary representation. Proof Every group element a is represented by a matrix D(a), which acts on the vector space. We denote the basis of the representation vector space by {e1 , . . . , ed }. We consider two vectors v and w: v=

d  i=1

vi e i ,

w=

d 

wi e i .

(A.3)

i=1

H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5, © Springer-Verlag Berlin Heidelberg 2012

229

230

A Useful Theorems

We define the scalar product between v and w as (v, w) =

d 

vi∗ wi .

(A.4)

i=1

Then we define another scalar product by v, w =

 1  D(a)v, D(a)w . NG

(A.5)

a∈G

It follows that   1  D(b)D(a)v, D(b)D(a)w D(b)v, D(b)w = NG



a∈G

=

 1  D(ba)v, D(ba)w NG a∈G

 1  D(c)v, D(c)w = NG c∈G

= v, w.

(A.6)

This implies that D(b) is unitary with respect to the scalar product v, w. The orthogonal bases {ei } and {ei } for the two scalar products (v, w) and v, w can be related by the linear transformation T such that ei = T ei , i.e., (v, w) = T v, T w. We define D  (g) = T −1 D(g)T . Then it follows that  −1    T D(a)T v, T −1 D(a)T w = D(a)T v, D(a)T w = T v, T w = (v, w). That is, the matrix D  (g) is unitary and is equivalent to D(g).

(A.7) 

Schur’s Lemma (I) Let D1 (g) and D2 (g) be irreducible representations of G, which are inequivalent to each other. If AD1 (g) = D2 (g)A,

∀g ∈ G,

(A.8)

the matrix A must vanish, i.e., A = 0. (II) If D(g)A = AD(g),

∀g ∈ G,

the matrix A must be proportional to the identity matrix I , i.e., A = λI .

(A.9)

A Useful Theorems

231

Proof (I) We denote the representation vector spaces for D1 (g) and D2 (g) by V and W , respectively. Consider a map A : V → W satisfying (A.8). We consider the kernel of A, viz., Ker(A) = {v ∈ V |Av = 0}.

(A.10)

Let v ∈ Ker(A). Then we have AD1 (g)v = D2 (g)Av = 0.

(A.11)

It follows that D1 (g)Ker(A) ⊂ Ker(A), that is, Ker(A) is invariant under D1 (g). Because D1 (g) is irreducible, this implies that Ker(A) = {0},

or

Ker(A) = V .

(A.12)

But Ker(A) = V implies that A = 0. On the other hand, if we consider the image Im(A) = {Av|v ∈ V },

(A.13)

D2 (g)Av = AD1 (g)v ∈ Im(A).

(A.14)

we find

That is, Im(A) is invariant under D2 (g). Once again, since D2 (g) is irreducible, this implies that Im(A) = {0},

or

Im(A) = W.

(A.15)

However, Im(A) = {0} means that A = 0. As a result, we find that A must satisfy A = 0,

or AD1 (g)A−1 = D2 (g).

(A.16)

The latter would mean that the representations D1 (g) and D2 (g) were equivalent to each other. Therefore, A must vanish, i.e., A = 0, if D1 (g) and D2 (g) are not equivalent. (II) Now we consider the case with D(g) = D1 (g) = D2 (g) and V = W . Hence, A is a linear operator on V . The finite-dimensional matrix A has at least one eigenvalue, because the characteristic equation det(A − λI ) = 0 has at least one root λ, and this λ is an eigenvalue. Then (A.9) leads to D(g)(A − λI ) = (A − λI )D(g),

∀g ∈ G.

(A.17)

Using the above proof of Schur’s lemma (I) and the fact that Ker(A − λI ) = {0}, we find Ker(A − λI ) = V , that is, A − λI = 0.  Theorem Let Dα (g) and Dβ (g) be irreducible representations of a group G on the

232

A Useful Theorems

dα and dβ dimensional vector spaces. Then they satisfy the orthogonality relation  a∈G

  NG Dα (a)i Dβ a −1 mj = δαβ δij δm . dα

(A.18)

Proof We define A=



  Dα (a)BDα a −1 ,

(A.19)

a∈G

where B is an arbitrary (dα × dα ) matrix. We find D(b)A = AD(b), since    Dα (b)Dα (a)BDα a −1 Dα (b)A = a∈G

=



a∈G

=



  Dα (ba)BDα (ba)−1 Dα (b)   Dα (c)BDα c−1 Dα (b).

(A.20)

c∈G

That is, by Schur’s lemma (II), we find that the matrix A must be proportional to the (dα × dα ) identity matrix. We choose Bij = δi δj m . Then we obtain Aij =



  Dα (a)i Dα a −1 mj ,

(A.21)

a∈G

and the right-hand side (RHS) can be written λ(, m)δij , that is, 

  Dα (a)i Dα a −1 mj = λ(, m)δij .

(A.22)

a∈G

Furthermore, we compute the trace of both sides. The trace of the RHS is computed to be λ(, m) tr δij = dα λ(, m),

(A.23)

while the trace of left-hand side (LHS) is obtained as d  

     Dα (a)i Dα a −1 mi = Dα aa −1 m

i=1 a∈G

a∈G

= NG δm . By comparing these results, we obtain λ(, m) =

NG δm . dα

(A.24)

A Useful Theorems

233

Hence, we find  a∈G

  NG Dα (a)i Dα a −1 mj = δij δm . dα

(A.25)

Similarly, we define A(αβ) =



  Dα (a)BDβ a −1 ,

(A.26)

a∈G

where Dα (a) and Dβ (a) are inequivalent. Then we find Dα (a)A = ADβ (a). Similarly to the previous analysis, using Schur’s lemma (I), we obtain    Dα (a)i Dβ a −1 mj = 0. (A.27) a∈G

Thus, we arrive at (A.18). Furthermore, if the representation is unitary, (A.18) becomes  NG Dα (a)i Dβ∗ (a)j m = δαβ δij δm . (A.28) dα a∈G

 Because of this orthogonality, we can expand an arbitrary function F (a) of a in terms of the matrix elements of irreducible representations:  α F (a) = cj,k Dα (a)j k . (A.29) α,j,k

Theorem The characters χα (g) and χβ (g) of representations of Dα (g) and Dβ (g) satisfy the orthogonality relation  χDα (g)∗ χDβ (g) = NG δαβ . (A.30) g∈G

Proof From (A.28), we obtain  g∈G

Dα (g)ii Dβ∗ (g)jj =

NG δαβ δij . dα

Thus, by summing over all i and j , we obtain (A.30). A class function is defined as any function F (a) of a which satisfies   F g −1 ag = F (a), ∀g ∈ G.

(A.31) 

(A.32)

Theorem The number of irreducible representations is equal to the number of conjugacy classes.

234

A Useful Theorems

Proof The class function can also be expanded in terms of the matrix elements of the irreducible representations as in (A.29). It then follows that F (a) =

1   −1  F g ag NG g∈G

=

  1  α cj,k Dα g −1 ag j k NG g∈G α,j,k

=

 1   α   −1  cj,k Dα g Dα (a)Dα (g) j k . NG

(A.33)

g∈G α,j,k

Using the orthogonality relation (A.28), we obtain F (a) =

 1 cα Dα (a) dα j,j

α,j,

 1 = cα χα (a). dα j,j

(A.34)

α,j

That is, any class function F (a), which is constant on conjugacy classes, can be expanded in terms of the characters χα (a). This implies that the number of irreducible representations is equal to the number of conjugacy classes.  Theorem The characters satisfy the orthogonality relation 

χDα (Ci )∗ χDα (Cj ) =

α

NG δC C , ni i j

(A.35)

where Ci and Cj denote the conjugacy classes and ni is the number of elements in the conjugacy class Ci . Proof We define the matrix Viα by  Viα =

ni χα (Ci ), NG

(A.36)

where ni is the number of elements in the conjugacy class Ci . Note that i and α label the conjugacy class Ci and the irreducible representation, respectively. The matrix Viα is a square matrix because the number of irreducible representations is equal to the number of conjugacy classes. Using Viα , the orthogonality relation (A.30) can be rewritten as V † V = 1, that is, V is unitary. Thus, we also obtain V V † = 1. This is the content of (A.35). 

References

235

References 1. Hamermesh, M.: Group Theory and Its Application to Physical Problems. Addison-Wesley, Reading (1962) 2. Georgi, H.: Front. Phys. 54, 1 (1982) 3. Ludl, P.O.: arXiv:0907.5587 [hep-ph] 4. Ramond, P.: Group Theory: A Physicist’s Survey. Cambridge University Press, Cambridge (2010)

Appendix B

Representations of S4 in Different Bases

For the group S4 , several bases of representations have been used in the literature. Most group-theoretical aspects such as conjugacy classes and characters are independent of the representation basis. Tensor products are also independent of the basis. For example, we always have 2 ⊗ 2 = 11 ⊕ 12 ⊕ 2,

(B.1)

in any basis. However, the component form of this equation is basis-dependent. For example, the singlets 11 and 12 on the RHS are represented by components of 2 on the LHS, but their forms depend on the representation basis, as we shall see below. For applications, it is useful to show the transformation of bases and tensor products explicitly for several bases. This is what we shall do below.

B.1 Basis I First, we examine the basis used in Sect. 3.2. All the elements of S4 can be written as products of the generators b1 and d4 , which satisfy (b1 )3 = (d4 )4 = e,

d4 (b1 )2 d4 = b1 ,

d4 b1 d4 = b1 (d4 )2 b1 .

These generators are represented on 2, 3, and 3 as follows:   ω 0 0 1 = , on 2, b1 = , d 4 1 0 0 ω2 ⎛ ⎛ ⎞ ⎞ 0 0 1 −1 0 0 b1 = ⎝ 1 0 0 ⎠ , d4 = ⎝ 0 0 −1 ⎠ , on 3, 0 1 0 0 1 0 ⎛ ⎛ ⎞ ⎞ 0 0 1 1 0 0 d4 = ⎝ 0 0 1 ⎠ , on 3 . b1 = ⎝ 1 0 0 ⎠ , 0 1 0 0 −1 0 H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5, © Springer-Verlag Berlin Heidelberg 2012

(B.2)

(B.3)

(B.4)

(B.5)

237

238

B Representations of S4 in Different Bases

The multiplication rules are then obtained as follows:  b1 a2 b2  ⊗ = (a1 b2 + a2 b1 )1 ⊕ (a1 b2 − a2 b1 )1 ⊕ , (B.6) b2 2 a1 b1 2 2 ⎞ ⎞ ⎛ ⎞ ⎛ ⎛  a1 b1 + a2 b1 a1 b1 − a2 b1 b1 a1 ⊗ ⎝ b2 ⎠ = ⎝ ω2 a1 b2 + ωa2 b2 ⎠ ⊕ ⎝ ω2 a1 b2 − ωa2 b2 ⎠ , (B.7) a2 2 b3 3 ωa1 b3 + ω2 a2 b3 3 ωa1 b3 − ω2 a2 b3 3 ⎛ ⎞ ⎛ ⎛ ⎞ ⎞  a1 b1 − a2 b1 a1 b1 + a2 b1 b1 a1 ⊗ ⎝ b2 ⎠ = ⎝ ω2 a1 b2 − ωa2 b2 ⎠ ⊕ ⎝ ω2 a1 b2 + ωa2 b2 ⎠ , (B.8) a2 2 b3 3 ωa1 b3 − ω2 a2 b3 3 ωa1 b3 + ω2 a2 b3 3 ⎛ ⎞ ⎛ ⎞  a1 b1 2 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b2 + a3 b3 )1 ⊕ a1 b1 + ωa2 2 b2 + ω a3 b3 a1 b1 + ω a2 b2 + ωa3 b3 2 a3 3 b3 3 ⎛ ⎛ ⎞ ⎞ a2 b3 + a3 b2 a2 b3 − a3 b2 ⊕ ⎝ a3 b1 + a1 b3 ⎠ ⊕ ⎝ a3 b1 − a1 b3 ⎠ , (B.9) a1 b2 + a2 b1 3 a1 b2 − a2 b1 3 ⎛ ⎞ ⎛ ⎞  a1 b1 2 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b2 + a3 b3 )1 ⊕ a1 b1 + ωa2 2 b2 + ω a3 b3 a1 b1 + ω a2 b2 + ωa3 b3 2 a3 3 b3 3 ⎞ ⎞ ⎛ ⎛ a2 b3 + a3 b2 a2 b3 − a3 b2 ⊕ ⎝ a3 b1 + a1 b3 ⎠ ⊕ ⎝ a3 b1 − a1 b3 ⎠ , (B.10) a1 b2 + a2 b1 3 a1 b2 − a2 b1 3 ⎛ ⎞ ⎛ ⎞  a1 b1 2 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b2 + a3 b3 )1 ⊕ a1 b1 + ωa22 b2 + ω a3 b3 −a1 b1 − ω a2 b2 − ωa3 b3 2 a3 3 b3 3 ⎞ ⎞ ⎛ ⎛ a2 b3 − a3 b2 a2 b3 + a3 b2 ⊕ ⎝ a3 b1 − a1 b3 ⎠ ⊕ ⎝ a3 b1 + a1 b3 ⎠ . (B.11) a1 b2 − a2 b1 3 a1 b2 + a2 b1 3 

a1 a2





B.2 Basis II Next we consider another basis, which is used, e.g., in [1]. Following this reference, we define the generators b1 and d4 by b = b1 and a = d4 . In this basis, the generators a and b are represented by  a=

−1 0 , 0 1

b=−

1 2



1 √ − 3

√ 3 , 1

on 2,

(B.12)

B.2 Basis II

239

⎛

−1 0 a=⎝ 0 0 0 1 ⎛ 1 0 a = ⎝0 0 0 −1

⎞ 0 −1 ⎠ , 0 ⎞ 0 1⎠, 0

⎛

⎞ 0 0 1 b = ⎝ 1 0 0 ⎠ , on 31 , 0 1 0 ⎛ ⎞ 0 0 1 b = ⎝ 1 0 0 ⎠ , on 32 , 0 1 0

(B.13)

(B.14)

where we define 31 ≡ 3 and 32 ≡ 3 hereafter. The generators a and b are represented in the real basis. On the other hand, the above generators b1 and d4 are represented in the complex basis. The bases for 2 transform by the unitary transformation U † gU , where  1 1 i U=√ . (B.15) 2 −1 i That is, the elements a and b are written in terms of b1 and d4 as 1 b = U b1 U = − 2 †



1 √ − 3

√ 3 , 1

 a = U d4 U = †

−1 0 , 0 1

(B.16)

in the real basis. For the triplets, the (b1 , d4 ) basis is the same as the (b, a) basis. The multiplication rules are thus obtained as follows: 



a1 a2

a1 a2







b ⊗ 1 b2 2 ⎛





a b + a2 b1 = (a1 b1 + a2 b2 )11 ⊕ (−a1 b2 + a2 b1 )12 ⊕ 1 2 a 1 b1 − a2 b2 2 ⎞

⎛

⎞

a1 a2



, 2

(B.17) ⎞

⎛

b1 √ a2 b 1 √ a1 b1 ( 3a1 b2 + a2 b2 ) ⎠ ⊕ ⎝ 21 ( √3a2 b2 − a1 b2 ) ⎠ , ⊗ ⎝ b2 ⎠ = ⎝ − 12 √ 1 2 b3 3 − 12 ( 3a2 b3 + a1 b3 ) 3 2 ( 3a1 b3 − a2 b3 ) 3 1 1





2

(B.18) ⎞

⎞ ⎛ ⎛ ⎞ a1 b 1 b1 √ √ a2 b1 ( 3a1 b2 + a2 b2 ) ⎠ , ⊗ ⎝ b2 ⎠ = ⎝ 21 ( √3a2 b2 − a1 b2 ) ⎠ ⊕ ⎝ − 12 √ 1 2 b3 3 − 12 ( 3a2 b3 + a1 b3 ) 3 2 ( 3a1 b3 − a2 b3 ) 3 ⎛

2

1

2

(B.19) 

⎛ ⎞ ⎞  b1 a1 √1 (a2 b2 − a3 b3 ) 2 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b2 + a3 b3 )11 ⊕ 1 √ (−2a1 b1 + a2 b2 + a3 b3 ) a3 3 b3 3 6 2 1 1 ⎛ ⎛ ⎞ ⎞ a2 b3 + a3 b2 a3 b2 − a2 b3 ⊕ ⎝ a1 b3 + a3 b1 ⎠ ⊕ ⎝ a1 b3 − a3 b1 ⎠ , (B.20) a1 b2 + a2 b1 3 a2 b1 − a1 b2 3 ⎛

1

2

240

B Representations of S4 in Different Bases

⎛ ⎞ ⎞   b1 a1 √1 (a2 b2 − a3 b3 ) 2 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b2 + a3 b3 )11 ⊕ 1 √ (−2a1 b1 + a2 b2 + a3 b3 ) a3 3 b3 3 6 2 2 2 ⎛ ⎛ ⎞ ⎞ a2 b3 + a3 b2 a3 b2 − a2 b3 ⊕ ⎝ a1 b3 + a3 b1 ⎠ ⊕ ⎝ a1 b3 − a3 b1 ⎠ , (B.21) a1 b2 + a2 b1 3 a2 b1 − a1 b2 3 1 2 ⎛ ⎞ ⎛ ⎞   a1 b1 √1 (2a1 b1 − a2 b2 − a3 b3 ) 6 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b2 + a3 b3 )12 ⊕ √1 (a2 b2 − a3 b3 ) a3 3 b3 3 2 2 1 2 ⎞ ⎞ ⎛ ⎛ a3 b2 − a2 b3 a2 b3 + a3 b2 ⊕ ⎝ a1 b3 − a3 b1 ⎠ ⊕ ⎝ a1 b3 + a3 b1 ⎠ . (B.22) a2 b1 − a1 b2 3 a1 b2 + a2 b1 3 ⎛

1

2

B.3 Basis III Next, we consider a different basis, which is used, e.g., in [2], with the generators s and t corresponding to d4 and b1 , respectively. These generators are represented as follows: 

 ω 0 0 1 s= , t= , on 2, 1 0 0 ω2 ⎛ ⎞ ⎛ −1 2ω 2ω2 1 0 1⎝ s= t = ⎝ 0 ω2 2ω 2ω2 −1 ⎠ , 3 0 0 2ω2 −1 2ω ⎛ ⎞ ⎛ 1 −2ω −2ω2 1 1⎝ t = ⎝0 s= −2ω −2ω2 1 ⎠, 3 0 −2ω2 1 −2ω

(B.23) ⎞ 0 0 ⎠, ω 0 ω2 0

on 31 , ⎞ 0 0 ⎠, ω

(B.24)

on 32 . (B.25)

The doublet of this basis [2] is the same as the (d4 , b1 ) basis. In the representations 31 and 32 , the (s, t) basis and (d4 , b1 ) basis are transformed by the unitary matrix ⎛ 1 1 1 ⎝ 1 ω Uω = √ 3 1 ω2

⎞ 1 ω2 ⎠ , ω

(B.26)

B.3 Basis III

241

which is the so-called magic matrix. That is, the elements s and t are obtained from d4 and b1 as follows: ⎛ ⎞ ⎞ ⎛ 1 0 0 −1 2ω 2ω2 1 s = Uω† d4 Uω = ⎝ 2ω 2ω2 −1 ⎠ , t = Uω† b1 Uω = ⎝ 0 ω2 0 ⎠ . 3 2 2 0 0 ω 2ω −1 2ω (B.27) For 32 , we also find s and t in the same way. The multiplication rules are then obtained as follows:    a1 b a b ⊗ 1 = (a1 b2 + a2 b1 )11 ⊕ (a1 b2 − a2 b1 )12 ⊕ 2 2 , (B.28) a2 2 b2 2 a1 b1 2 ⎞ ⎞ ⎛ ⎞ ⎛ ⎛  b1 a1 b2 + a2 b3 a1 b2 − a2 b3 a1 ⊗ ⎝ b2 ⎠ = ⎝ a1 b3 + a2 b1 ⎠ ⊕ ⎝ a1 b3 − a2 b1 ⎠ , (B.29) a2 2 b3 3 a1 b1 + a2 b2 3 a1 b1 − a2 b2 3 1 1 2 ⎛ ⎞ ⎛ ⎛ ⎞ ⎞  b1 a1 b2 − a2 b3 a1 b2 + a2 b3 a1 ⊗ ⎝ b2 ⎠ = ⎝ a1 b3 − a2 b1 ⎠ ⊕ ⎝ a1 b3 + a2 b1 ⎠ , (B.30) a2 2 b3 3 a1 b1 − a2 b2 3 a1 b1 + a2 b2 3 2 1 2 ⎛ ⎞ ⎛ ⎞  a1 b1 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b3 + a3 b2 )11 ⊕ a2 b2 + a1 b3 + a3 b1 a3 b3 + a1 b2 + a2 b1 2 a3 3 b3 3 1 1 ⎞ ⎞ ⎛ ⎛ 2a1 b1 − a2 b3 − a3 b2 a2 b3 − a3 b2 ⊕ ⎝ 2a3 b3 − a1 b2 − a2 b1 ⎠ ⊕ ⎝ a1 b2 − a2 b1 ⎠ , 2a2 b2 − a1 b3 − a3 b1 3 a3 b1 − a1 b3 3 1

2

(B.31)

⎛

⎞ ⎛ ⎞  a1 b1 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b3 + a3 b2 )11 ⊕ a2 b2 + a1 b3 + a3 b1 a3 b3 + a1 b2 + a2 b1 2 a3 3 b3 3 2 2 ⎛ ⎛ ⎞ ⎞ 2a1 b1 − a2 b3 − a3 b2 a2 b3 − a3 b2 ⊕ ⎝ 2a3 b3 − a1 b2 − a2 b1 ⎠ ⊕ ⎝ a1 b2 − a2 b1 ⎠ , 2a2 b2 − a1 b3 − a3 b1 3 a3 b1 − a1 b3 3 1

⎛

⎞

⎛

2

(B.32)

⎞

a1 b1 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b3 + a3 b2 )12 ⊕ a2 b2 + a1 b3 + a3 b1 −a3 b3 − a1 b2 − a2 b1 2 a3 3 b3 3 1 2 ⎛ ⎛ ⎞ ⎞ a2 b3 − a3 b2 2a1 b1 − a2 b3 − a3 b2 ⊕ ⎝ a1 b2 − a2 b1 ⎠ ⊕ ⎝ 2a3 b3 − a1 b2 − a2 b1 ⎠ . a3 b1 − a1 b3 3 2a2 b2 − a1 b3 − a3 b1 3 

1

2

(B.33)

242

B Representations of S4 in Different Bases

B.4 Basis IV Here we consider another basis, which is used, e.g., in [3], with the generators t˜ and s˜ satisfying t˜4 = s˜ 2 = (˜s t˜)3 = (t˜s˜ )3 = e. (B.34) These generators are represented by √   1 −1 3 1 0 √ , t˜ = , s˜ = 0 −1 3 1 2 ⎛

⎛

⎞ −1 0 0 t˜ = ⎝ 0 −i 0 ⎠ , 0 0 i ⎛

0

⎜ 1 s˜ t˜ = ⎝ √2 √1 2

√i

2 − 2i i 2

on 31 , and ⎛ ⎞ 1 0 0 t˜ = ⎝ 0 i 0 ⎠ , 0 0 −i

− √i

0 ⎜ − √1 s˜ = ⎝ 2 − √1 2

⎞

s˜ t˜ =

− √1 1 2

2

1 2

− √1



−1 √ 3

√ − 3 , −1

on 2, (B.35)

⎞

2

− 12 ⎟ ⎠,

− 12

1 2

2

− 2i ⎟ ⎠,

(B.36)

i 2

⎛

0

⎜ 1 s˜ = ⎝ √2 √1 2

√1 2 − 12 1 2

√1 2 1 2 − 12

⎞

⎛

⎟ ⎠,

0

⎜ 1 s˜ t˜ = ⎝ √2 √1 2

√i

2 − 2i i 2

− √i

⎞

2

− 2i ⎟ ⎠, i 2

(B.37) on 32 . For the representation 2, the unitary transformation matrix Udoublet given by  1 1 i (B.38) Udoublet = √ 2 1 −i is used and the elements t˜ and s˜ t˜ are given in terms of d1 and b1 by  1 0 † ˜t = Udoublet d4 Udoublet = , 0 −1 √  1 −1 − 3 † √ b1 Udoublet = . s˜ t˜ = Udoublet 3 −1 2

(B.39)

On the other hand, for the representations 31 and 32 , the unitary transformation matrix Utriplet given by ⎛ ⎞ 1 0 0 1 ⎜ √1 ⎟ Utriplet = ⎝ 0 √2 (B.40) 2 ⎠ i i √ √ 0 − 2

is used.

2

B.4 Basis IV

243

For 31 , the elements t˜ and s˜ t˜ are given in terms of d4 and b1 by ⎛ ⎞ −1 0 0 † d4 Utriplet = ⎝ 0 −i 0 ⎠ , t˜ = Utriplet 0 0 i ⎞ ⎛ √i 0 − √i 2 2 ⎜ 1 i i ⎟ † s˜ t˜ = Utriplet b1 Utriplet = ⎝ √2 − 2 − 2 ⎠ . i 2

√1 2

(B.41)

i 2

For 32 , we find the same transformations. The multiplication rules are as follows:    a1 b1 a2 b2 − a1 b1 ⊗ = (a1 b1 + a2 b2 )11 ⊕ (a1 b2 − a2 b1 )12 ⊕ , a2 2 b2 2 a1 b2 + a2 b1 2 

a1 a2



⎛

⎞

⎛

⎞

a1 a2

⎛



⎞

⎛

⎞

(B.42)

⎞

⎛

a1 b1 −a2 b1 b1 √ √ ⎟ ⎟ ⎜ 3 ⎜ 3 1 ⎝ ⎠ b a b − a b a ⊗ = ⎝ √2 2 3 2 1 2 ⎠ ⊕ ⎝ √2 1 b3 + 12 a2 b2 ⎠ , (B.43) 2 2 3 3 b3 3 a2 b2 − 1 a1 b3 a1 b2 + 1 a2 b3 1



⎞

⎛

⎛

2

2

31

⎞

⎛

2

2

32

⎞

−a2 b1 a1 b1 b1 √ √ ⎟ ⎟ ⎜ ⎜ 3 3 1 ⊗ ⎝ b2 ⎠ = ⎝ √2 a1 b3 + 2 a2 b2 ⎠ ⊕ ⎝ √2 a2 b3 − 12 a1 b2 ⎠ , (B.44) 2 3 3 b3 3 a1 b2 + 1 a2 b3 a2 b2 − 1 a1 b3 2

2

2

31

2

2

32

 a1 b1 − 12 (a2 b3 + a3 b2 ) a1 b√ 1 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b3 + a3 b2 )11 ⊕ 3 2 (a2 b2 + a3 b3 ) a3 3 b3 3 2 1 1 ⎞ ⎞ ⎛ ⎛ a3 b3 − a2 b2 a3 b2 − a2 b3 ⊕ ⎝ a1 b3 + a3 b1 ⎠ ⊕ ⎝ a2 b1 − a1 b2 ⎠ , (B.45) −a1 b2 − a2 b1 3 a1 b3 − a3 b1 3 1 2 ⎛ ⎞ ⎛ ⎞   b1 a1 a1 b√ − 12 (a2 b3 + a3 b2 ) 1 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b3 + a3 b2 )11 ⊕ 3 2 (a2 b2 + a3 b3 ) a b 3

⎛

32

3

32



2

⎛

⎛

⎞

⎞

a3 b3 − a2 b2 a3 b2 − a2 b3 ⊕ ⎝ a1 b3 + a3 b1 ⎠ ⊕ ⎝ a2 b1 − a1 b2 ⎠ , −a1 b2 − a2 b1 3 a1 b3 − a3 b1 3 1

(B.46)

2

⎛ ⎞ ⎞ a1 b1 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b3 + a3 b2 )12 a3 3 b3 3 1 2  √ ⊕ 23 (a2 b2 + a3 b3 ) − a1 b1 + 12 (a2 b3 + a3 b2 )

2

244

B Representations of S4 in Different Bases

⎛

⎛ ⎞ ⎞ a3 b2 − a2 b3 a3 b3 − a2 b2 ⊕ ⎝ a2 b1 − a1 b2 ⎠ ⊕ ⎝ a1 b3 + a3 b1 ⎠ . a1 b3 − a3 b1 3 −a1 b2 − a2 b1 3 1

(B.47)

2

References 1. Hagedorn, C., Lindner, M., Mohapatra, R.N.: J. High Energy Phys. 0606, 042 (2006). arXiv:hep-ph/0602244 2. Bazzocchi, F., Merlo, L., Morisi, S.: Nucl. Phys. B 816, 204 (2009). arXiv:0901.2086 [hep-ph] 3. Altarelli, G., Feruglio, F., Merlo, L.: J. High Energy Phys. 0905, 020 (2009). arXiv:0903.1940 [hep-ph]

Appendix C

Representations of A4 in Different Bases

C.1 Basis I In Sect. C.2, we discuss another basis for representation of the A4 group. But first consider the basis used in Sect. 4.1. All elements of A4 can be written as products of the generators s and t , which satisfy s 2 = t 3 = (st)3 = e. On the representation 3, these generators are represented by ⎛ ⎛ ⎞ ⎞ 1 0 0 0 0 1 s = a2 = ⎝ 0 −1 0 ⎠ , t = b1 = ⎝ 1 0 0 ⎠ . 0 0 −1 0 1 0

(C.1)

(C.2)

The multiplication rule of the triplet is thus ⎛ ⎞ ⎛ ⎞ b1 a1   ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b2 + a3 b3 )1 ⊕ a1 b1 + ωa2 b2 + ω2 a3 b3  1 a3 3 b3 3   ⊕ a1 b1 + ω2 a2 b2 + ωa3 b3 1 ⎞ ⎞ ⎛ ⎛ a2 b3 + a3 b2 a2 b3 − a3 b2 ⊕ ⎝ a3 b1 + a1 b3 ⎠ ⊕ ⎝ a3 b1 − a1 b3 ⎠ . (C.3) a1 b2 + a2 b1 3 a1 b2 − a2 b1 3

C.2 Basis II Next we consider another basis, used, e.g., in [1]. In this basis, we denote the generators by a and b, which correspond to s and t , respectively, and these generators H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5, © Springer-Verlag Berlin Heidelberg 2012

245

246

C Representations of A4 in Different Bases

are represented as follows: ⎛ ⎞ −1 2 2 1 a = ⎝ 2 −1 2 ⎠ , 3 2 2 −1

⎛

1 0 b = ⎝ 0 ω2 0 0

⎞ 0 0 ⎠, ω

(C.4)

on the representation 3. These bases are transformed by the unitary transformation matrix Uω given by ⎞ ⎛ 1 1 1 1 ⎝ 1 ω ω2 ⎠ , (C.5) Uω = √ 3 1 ω2 ω and the elements a and b are written as ⎛ ⎞ −1 2 2 1⎝ † 2 −1 2 ⎠ , a = Uω sUω = 3 2 2 −1

⎛

b

= Uω† tUω

1 0 ⎝ = 0 ω2 0 0

⎞ 0 0 ⎠ . (C.6) ω

Therefore, the multiplication rule of the triplet is ⎛ ⎞ ⎛ ⎞ a1 b1 ⎝ a2 ⎠ ⊗ ⎝ b2 ⎠ = (a1 b1 + a2 b3 + a3 b2 )1 ⊕ (a3 b3 + a1 b2 + a2 b1 )1 a3 3 b3 3 ⊕ (a2 b2 + a1 b3 + a3 b1 )1 ⎛ ⎛ ⎞ ⎞ 2a b − a2 b3 − a3 b2 a2 b3 − a3 b2 1⎝ 1 1 1 2a3 b3 − a1 b2 − a2 b1 ⎠ ⊕ ⎝ a1 b2 − a2 b1 ⎠ . ⊕ 3 2a b − a b − a b 2 a b −a b 2 2

1 3

3 1

3

1 3

3 1

3

(C.7)

References 1. Altarelli, G., Feruglio, F.: Nucl. Phys. B 741, 215 (2006). arXiv:hep-ph/0512103

Appendix D

Representations of A5 in Different Bases

D.1 Basis I In Sect. D.2, we show another basis for representations of the A5 group, but first, let us discuss the basis used in Sect. 4.2. All the elements of A5 can be written as products of the generators s and t , which satisfy 3  s 2 = t 5 = t 2 st 3 st −1 stst −1 = e.

(D.1)

The generators s and t are represented as follows [1]: ⎛

−1 1⎜ s= ⎝ φ 2 1 ⎛ 1⎜ s= ⎝ 2

φ

−φ

⎛

1 φ

1

φ

1 φ

⎞

⎟ 1 ⎠, 1 −φ ⎞ 1 1 φ ⎟ −1 φ ⎠ , φ φ1 1 φ

−1 −1 −3 1⎜ −1 3 1 s= ⎜ 1 −1 4 ⎝ −3 √ √ √ 5 − 5 − 5 ⎛ −1 1 −3 1⎜ −1 −3 1 t= ⎜ 1 1 4 ⎝ √3 √ √ 5 − 5 − 5

⎛

1 1⎜ t = ⎝ −φ 2 1 ⎛ 1⎜ t= ⎝ 2 √ ⎞ −√5 −√ 5 ⎟ ⎟, 5 ⎠ −1 √ ⎞ √5 ⎟ √5 ⎟ , 5⎠ −1

φ

−φ 1 φ

−1

φ 1 φ

−1 − φ1 1 φ

1 φ

⎞

⎟ 1 ⎠ , on 3, (D.2) φ ⎞ 1 ⎟ φ ⎠ , on 3 , (D.3) − φ1

(D.4) on 4,

H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5, © Springer-Verlag Berlin Heidelberg 2012

247

248

D Representations of A5 in Different Bases

⎛

s=

1 2

1−3φ ⎜ 42 ⎜ φ ⎜ 2 ⎜ ⎜− 1 ⎜ √2φ 2 ⎜ ⎜ 5 ⎝ √2 3 4φ

⎛

t=

1 2

1−3φ ⎜ 42 ⎜ φ ⎜ 2 ⎜ ⎜ 1 ⎜ 2φ√2 ⎜ ⎜− 5 ⎝ √2 3 4φ

φ2 2

√

− 2φ1 2

5 2

√

⎞

3 4φ ⎟ √ 3 ⎟ 1 1 0 ⎟ 2φ √ ⎟ 3φ ⎟ , 1 0 −1 − 2 ⎟ √ ⎟ 0 −1 1√ − 23 ⎟ ⎠ √ √ 3φ−1 3φ 3 3 − − 2φ 2 2 4 √ ⎞ √ 2 3 − φ2 − 2φ1 2 − 25 4φ ⎟ √ 3 ⎟ −1 1 0 ⎟ √2φ ⎟ 3φ ⎟ , 1 0 −1 ⎟ √2 ⎟ 3 ⎟ 0√ 1 1 2 ⎠ √ √ 3φ 3φ−1 3 3 − 2φ − 2 2 4

(D.5)

√ on 5, where φ = (1 + 5)/2. The tensor products decompose as follows: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x1 y1 x3 y2 − x2 y3 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = (x1 y1 + x2 y2 + x3 y3 )1 ⊕ ⎝ x1 y3 − x3 y1 ⎠ x3 3 y3 3 x2 y1 − x1 y2 3 ⎞ ⎛ x2 y 2 − x 1 y 1 ⎟ ⎜ x 2 y 1 + x1 y 2 ⎟ ⎜ ⎟ , ⎜ x y + x y 3 2 2 3 ⊕⎜ ⎟ ⎠ ⎝ x 1 y 3 + x3 y 1 1 √ − (x1 y1 + x2 y2 − 2x3 y3 ) 3

(D.6)

5

⎛

⎞ ⎞ ⎛ ⎞ ⎛ x1 y1 x3 y2 − x2 y3 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = (x1 y1 + x2 y2 + x3 y3 )1 ⊕ ⎝ x1 y3 − x3 y1 ⎠ x3 3 y3 3 x2 y1 − x1 y2 3 ⎞ ⎛ √   1 1 2 − φ x1 y1 − φx2 y2 + 5x3 y3 ⎟ ⎜ x2 y1 + x1 y2 ⎟ ⎜ ⎟ ⎜ −(x3 y1 + x1 y3 ) ⊕⎜ ⎟ , ⎟ ⎜ y + x y x 2 3 3 2 ⎝  ⎠ 1 √ (1 − 3φ)x1 y1 + (3φ − 2)x2 y2 + x3 y3 2 3

5

(D.7) ⎞ − φx1 y3 x1 y1 ⎟ ⎜ φx3 y1 + φ1 x2 y3 ⎟ ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎜ ⎟ ⎜ 1 x3 3 y3 3 ⎝ − φ x1 y1 + φx2 y2 ⎠ x2 y1 − x1 y2 + x3 y3 4 ⎛

⎞

⎛

⎞

⎛

1 φ x3 y2

D.1 Basis I

249

√ ⎞ ⎛ 1 2 1 2 φ x2 y1 + φ 2 x1 y2 − 5x3 y3 ⎟ ⎜ − φx1 y1 + φ1 x2 y2 ⎟ ⎜ ⎟ ⎜ 1 ⎟ , x y − φx y ⊕⎜ 3 1 2 3 φ ⎟ ⎜ 1 ⎟ ⎜ φx3 y2 + φ x1 y3 ⎠ ⎝ √   3 1 2 φ x2 y1 + φx1 y2 + x3 y3

(D.8)

5

⎞

⎛

⎞ ⎛ 1 ⎞ y1 − φ 2 x1 y3 + φ1 x2 y4 + x3 y2 x1 ⎜ y2 ⎟ ⎟ 1 1 ⎝ x2 ⎠ ⊗ ⎜ ⎟ = ⎜ ⎝ − φ x1 y4 + x2 y3 + φ 2 x3 y1 ⎠ ⎝ y3 ⎠ x3 3 −x1 y1 + φ12 x2 y2 + φ1 x3 y4  y4 4 3 ⎞ ⎛ −x1 y3 + x2 y4 − x3 y2 ⎜ −x1 y4 − x2 y3 + x3 y1 ⎟ ⎟ ⎜ ⊕⎝ x1 y1 + x2 y2 + x3 y4 ⎠ x1 y2 − x2 y1 − x3 y3 4 ⎞ ⎛ 1 2 (6φ + 5)x1 y2 + (3φ + 4)x2 y1 + (3φ + 1)x3 y3 ⎜ ⎟ −x1 y1 + (3φ + 2)x2 y2 − (3φ + 1)x3 y4 ⎜ ⎟ ⎜ ⎟ , −(3φ + 1)x1 y4 − x2 y3 − (3φ + 2)x3 y1 ⊕⎜ ⎟ ⎝ ⎠ −(3φ + 2)x1 y3 − (3φ + 1)x2 y4 + x3 y2 √   3 2 x1 y2 − (3φ + 2)x2 y1 + 3(φ + 1)x3 y3 5 ⎛

(D.9) ⎞ ⎛ ⎞ y1 x1 y3 + φx2 y4 + φ 2 x3 y1 x1 ⎜ y2 ⎟ ⎝ x2 ⎠ ⊗ ⎜ ⎟ = ⎝ −φx1 y4 − φ 2 x2 y3 − x3 y2 ⎠ ⎝ y3 ⎠ x3 3 −φ 2 x1 y2 − x2 y1 − φx3 y4 3 y4 4 ⎛ ⎞ x1 y4 − x2 y3 + x3 y2 ⎜ x1 y3 + x2 y4 − x3 y1 ⎟ ⎟ ⊕⎜ ⎝ −x1 y2 + x2 y1 + x3 y4 ⎠ −(x1 y1 + x2 y2 + x3 y3 ) 4 ⎛ ⎞ x1 y1 − φ 4 x2 y2 + φ 2 (2φ − 1)x3 y3 ⎜ x1 y2 − φ 4 x2 y1 + φ 2 (2φ − 1)x3 y4 ⎟ ⎜ 4 ⎟ 2 ⎟ ⊕⎜ ⎜ φ x1 y3 − φ (2φ − 1)x2 y4 + x3 y1 ⎟ , (D.10) 2 4 ⎝ φ (2φ − 1)x1 y4 − x2 y3 − φ x3 y2 ⎠ √   − 3φ φ 2 x1 y1 − x2 y2 − φx3 y3 5 ⎛

⎞

⎛

⎞  y1 ⎛  ⎞ x1 y1 + √1 y5 − x2 y2 − x3 y4 ⎜ y2 ⎟ x1 3   ⎜ ⎟ ⎟ 1 ⎝ x2 ⎠ ⊗ ⎜ y 3 ⎟ = ⎜ ⎝ −x1 y2 − x2 y1 − √3 y5 − x3 y3 ⎠ ⎜ ⎟ x3 3 ⎝ y 4 ⎠ −x1 y4 − x2 y3 − √2 x3 y5 3 3 y5 5 ⎛

⎞

⎛

250

D Representations of A5 in Different Bases √ ⎞ 3 1 2 x2 y5 − φ 2 x3 y3 2φ ⎟ ⎜ √3 1 1 ⎟ ⊕⎜ ⎝ − 2 x1 y5 − 2φ 3 x1 y1√+ φ 2 x2 y2√− x3 y4 ⎠ − φ12 x1 y4 + x2 y3 + 2φ5 x3 y1 − 2φ3 x3 y5  3 √ ⎞ ⎛ 1 2 x y + φ 2−6 x2 y1 + 23 φ 2 x2 y5 + φ 2 x3 y3 φ2 1 2 √ ⎟ ⎜ φ+4 ⎜− x1 y1 − 2φ32 x1 y5 − φ 2 x2 y2 − φ12 x3 y4 ⎟ 2 ⎟ ⎜ ⊕⎜ √ √ ⎟ ⎝ φ 2 x1 y4 + 12 x2 y3 − 25 x3 y1 − 3 2 3 x3 y5 ⎠ √φ

⎛

x1 y2 − φ2 x2 y1 −

5(x1 y3 + x2 y4 + x3 y2 ) ⎞ x1 y3 + x2 y4 − 2x3 y2 ⎟ ⎜ x1 y4 − x2 y3 + 2x3 y√1 ⎟ ⎜ ⎟ −x y + x y − x y + 3x y ⊕⎜ 2 2 3 4 ⎜ 1 1 √ 1 5⎟ , ⎝ −x1 y2 − x2 y1 + x3 y3 − 3x2 y5 ⎠ √ − 3(x1 y3 − x2 y4 ) 5

4

⎛

(D.11)

⎛

⎞ √ y1 ⎞ ⎛ 1 ⎜ y2 ⎟ x2 y1 + 23 φ 2 x2 y5 + x3 y4 −φ 2 x1 y2 + 2φ x1 ⎜ ⎟ √ ⎟ 3 ⎝ x2 ⎠ ⊗ ⎜ y3 ⎟ = ⎜ 2 ⎠ ⎝ 2φ+1 x y + 1 1 ⎜ ⎟ 2 2 x1√y5 − x2 y2 −√φ x3 y3 x3 3 ⎝ y4 ⎠ 5 3 2 x1 y3 + φ x2 y4 − 2 φx3 y1 + 2 φx3 y5 3 y5 5 ⎛ 1 ⎞ 3φ−1 √ 2φ x1 y1 − x2 y2 + x3 y3 + 2 3 x1 y5 ⎜ ⎟ √ x y ⎟ −x y + φ x y − x y − 3φ−2 ⊕⎜ 2 3 2 5⎠ ⎝ 1 2 2 2 1√ 3 4 1 x1 y3 − x2 y4 − 25 x3 y1 − √ x y 2 3 3 5 3 ⎛ √   ⎞ 1 1 1 2 √ 3φx1 y5 2 x1 y1 + φ x2 y2 + φ 2 x3 y3 − φ √ ⎜ 1 5 1 ⎟ ⎜ √ − x1 y2 − φ 2 x2 y1 − 3(φ−3) √ x2 y5 − φ 2 x3 y4 ⎟ 2 φ 5 5 ⊕⎜ √ √   ⎟ ⎟ ⎜ ⎠ ⎝ √1 φ 2 x1 y3 − φ12 x2 y4 + 5x3 y1 + 3x3 y5 5 x1 y4 − x2 y3 + x3 y2 4 ⎛ ⎞ −(3φ − 1)x1 y4 + (2 − 3φ)x2√y3 + x3 y2 ⎜ ⎟ −2x1 y3 − 2x2 y4 − x3 y1 + 15x √ 3 y5 ⎜ ⎟ ⎜ 2x1 y2 − (2 − 3φ)x2 y1 − 2x3 y4 + 3φx2 y5 ⎟ ⊕⎜ ⎟ ,(D.12) √ ⎜ (3φ − 1)x y + 2x y + 2x y − 3 x y ⎟ 1 1 2 2 3 3 ⎝ φ 1 5 ⎠ √ √ √ 3 x y − φ 3x y − 15x y 2 3 3 2 φ 1 4 ⎛

⎞

5

⎞ ⎛ ⎞ x1 y1 ⎜ x2 ⎟ ⎜ y2 ⎟ ⎜ ⎟ ⊗ ⎜ ⎟ = (x1 y1 + x2 y2 + x3 y3 + x4 y4 )1 ⎝ x3 ⎠ ⎝ y3 ⎠ x4 4 y4 4 ⎛

D.1 Basis I

251

⎞ x1 y3 + x2 y4 − x3 y1 − x4 y2 ⊕ ⎝ −x1 y4 + x2 y3 − x3 y2 + x4 y1 ⎠ x1 y2 − x2 y1 − x3 y4 + x4 y3 3 ⎛

⎛

⎞ x1 y4 + x2 y3 − x3 y2 − x4 y1 ⊕ ⎝ −x1 y3 + x2 y4 + x3 y1 − x4 y2 ⎠ x1 y2 − x2 y1 + x3 y4 − x4 y3 3 √ √ ⎞ 5x3 y2 + x4 y1 x√ 1 y4 − 5x2 y3 − √ ⎜ − 5x1 y3 + x2 y4 − 5x3 y1 + x4 y2 ⎟ ⎟ √ √ ⊕⎜ ⎝ − 5x1 y2 − 5x2 y1 + x3 y4 + x4 y3 ⎠ x1 y1 + x2 y2 + x3 y3 − 3x4 y4 4 ⎛ ⎞ 2 φ 1 − √ x1 y1 + √ 2 x2 y2 + x3 y3 5φ ⎜ 1 5 ⎟ ⎜ − √ x1 y2 − √1 x2 y1 − x3 y4 − x4 y3 ⎟ ⎜ ⎟ 5 5 ⎜ 1 ⎟ 1 ⊕ ⎜ √5 x1 y3 + x2 y4 + √5 x3 y1 + x4 y2 ⎟ , ⎜ ⎟ ⎜ −x1 y4 − √1 x2 y3 − √1 x3 y2 − x4 y1 ⎟ ⎝ ⎠ 5 5    3 1 − 5 φ x1 y1 − φx2 y2 + x3 y3 ⎛

⎛

⎛

⎞

5

⎞ y1 x1 ⎜ y2 ⎟ ⎜ ⎟ ⎜ x2 ⎟ ⎜ ⎟ ⊗ ⎜ y3 ⎟ ⎜ ⎟ ⎝ x3 ⎠ ⎝ y4 ⎠ x4 4 y5 5 ⎞ ⎛ √ √ 2 2 x y + 2 5x y − 3 x y x y − (φ + 4)x y + 2φ 1 2 2 1 3 4 4 3 2 5 2 2 φ√ ⎟ ⎜ φ √ 2 2 2 ⎟ =⎜ 5 − 2φ x2 y2 + φ 2 x3 y3 + 2 5x4 y4 ⎠ ⎝ (φ − 5)x1 y1 + 3φ x1 y√ √ √ 2 2 2φ x1 y3 − φ 2 x2 y4 − 5x3 y1 − 3 3x3 y5 + 2 5x4 y2 3 ⎞ ⎛ √ √ 1 1 2 3φx1 y5 − φ 2 x2 y2 + φ x3 y3 + 5x4 y4 2 x1 y1 − √ ⎟ ⎜φ 2 √ 3 1 2 ⎟ ⊕⎜ ⎝ −φ x1 y2 − φ x2 y1 +√ φ x2 y5 −√φ 2 x3 y4 −√ 5x4 y3 ⎠ 1 2 x y − φ x2 y4 + 5x3 y1 + 3x3 y5 + 5x4 y2  φ2 1 3 3 √ ⎞ 2 φ 3 1 1 − √ x1 y1 − √ x2 y2 + √ x3 y3 − x4 y4 − √ x1 y5 5 5 5φ 5 ⎟ ⎜ ⎟ ⎜ √1 1 3 √1 x3 y4 + x4 y3 ⎟ x y + φx y − ⎜ − 5 x1 y2 + √5φ 2 1 2 5 2 5 5 ⊕⎜ ⎟  ⎟ ⎜ 3 √1 x1 y3 − √1 x2 y4 + x3 y1 − ⎠ ⎝ x y − x y 3 5 4 2 5 5 5 −x1 y4 + x2 y3 − x3 y2 4

⎛

(D.13)

252

D Representations of A5 in Different Bases

⎛ ⎜ ⎜ ⎜ ⎜ ⊕⎜ ⎜ ⎜ ⎜ ⎝ ⎛

1 2

1 √ 2 3





 2 φ x1 y4 −

1 x y − 3x3 y2 + 3x4 y1 + 53 x4 y5 φ2 2 3  φx1 y3 − φ1 x2 y4 − x3 y1 − x4 y2 + 53 x3 y5 1 1 √1 2 φ x1 y2 + φ x2 y1 + 3 φ x2 y5 + φx3 y4 − x4 y3 φx1 y1 + √ 1 2 x1 y5 − φx2 y2 + φ1 x3 y3 − x4 y4 3φ

−(φ − 5)x1 y4 + (φ + 4)x2 y3 +

√

5x3 y2 −

√

√

⎞



 5x4 y1 + 32 x4 y5

⎟ ⎟ ⎟ ⎟ ⎟ , ⎟ ⎟ ⎟ ⎠

⎞

x1 y4 − x2 y3 − 2x3 y2 + 32 x4 y√1 + 215 x4 y5 ⎟ ⎜ ⎜ φ 2 x1 y3 + 12 x2 y4 − 12 x3 y1 + 215 x3 y5 − x4 y2 ⎟ φ ⎟ ⎜ √ ⎜ 3 2x y − x y ⎟ , ⊕ ⎜ − φ12 x1 y2 + 3φ−2 x y + φx y + φ ⎟ 2 1 2 5 3 4 4 3 2 2 ⎟ ⎜ √ ⎜ 3φ−1 x y − φ 2 x y − 1 x y − x y − 3 x y ⎟ 1 1 2 2 4 4 1 5 ⎠ ⎝ 2 2φ φ2 3 3 √ √ √ 3x1 y4 + 3x2 y3 + 32 x4 y5 − 215 x4 y1 5 ⎛ ⎞ ⎛ ⎞ x1 y1 ⎜ x2 ⎟ ⎜ y2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x3 ⎟ ⊗ ⎜ y3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ x4 ⎠ ⎝ y4 ⎠ x5 5 y5 5

(D.14)

= (x1 y1 + x2 y2 + x3 y3 + x4 y4 + x5 y5 )1 √ ⎞ ⎛ x1 y3 − x3 y1 + x2 y4 − x4 y2 + √3(x3 y5 − x5 y3 ) ⊕ ⎝ x1 y4 − x4 y1 − (x2 y3 − x3 y2 ) − 3(x4 y5 − x5 y4 ) ⎠ −2(x1 y2 − x2 y1 ) − (x3 y4 − x4 y3 ) 3 √ ⎛ ⎞ (2φ + 3)(x1 y4 − x4 y1 ) + 2φ(x2 y3 − x3 y2 ) + 4 y5 − x5 y4 ) √ 3(x 2 ⎠ ⊕ ⎝ (φ + 3)(x1 y3 − x3 y1 ) + √2φ(x2 y4 − x4 y2 ) − 3φ (x3 y5 − x5 y3 ) −φ(x1 y2 − x2 y1 ) − 15φ(x2 y5 − x5 y2 ) + 2φ(x3 y4 − x4 y3 ) 3 ⎞ ⎛ √  2 √  5 φ (x1 y4 + x4 y1 ) + x2 y3 + x3 y2 + 2φ3 (x4 y5 + x5 y4 ) √ ⎟ ⎜√  2 ⎜ 5 1 (x1 y3 + x3 y1 ) − (x2 y4 + x4 y2 ) + 3 φ(x3 y5 + x5 y3 ) ⎟ 2 ⎟ ⎜ 2 ⊕ ⎜ √  2φ√ √ ⎟ ⎝ 5 − 5(x1 y2 + x2 y1 ) + 3(x2 y5 + x5 y2 ) + 2(x3 y4 + x4 y3 ) ⎠ 2 3x1 y1 − 2(x2 y2 + x3 y3 + x4 y4 ) + 3x5 y5 4 ⎛ 1 1 ⎞ 2 φ √ √ 2φ (x1 y4 − x4 y1 ) − (x2 y3 − x3 y2 ) + 2 3 (x4 y5 − x5 y4 ) ⎟ ⎜ 1 5 φ 1 ⎜ √ 2 (x1 y3 − x3 y1 ) − (x2 y4 − x4 y2 ) + √ 2 (x3 y5 − x5 y3 ) ⎟ ⎟ 5 2 3φ ⊕⎜ ⎜ √ ⎟ 1 1 1 ⎝ 2 5 (x1 y2 − x2 y1 ) − 2√3 (x2 y5 − x5 y2 ) − √5 (x3 y4 − x4 y3 ) ⎠ − √1 (x1 y5 − x5 y1 ) 3

5

4

D.2 Basis II ⎛

⊕

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

253

√  ⎞  11 −x1 y1 − √ (x y + x5 y1 ) + 43 x2 y2 − 4155 φx3 y3 + φ1 x4 y4 + x5 y5 3 15 1 5   ⎟ 4 √1 √2 ⎟ 3 x1 y2 + x2 y1 + 15 (x2 y5 + x5 y2 ) + 5 (x3 y4 + x4 y3 ) ⎟   ⎟ 4 √ √ (x3 y5 + x5 y3 ) −φ(x1 y3 + x3 y1 ) + 2(x2 y4 + x4 y2 ) − 2−3φ ⎟ , 3 5 3 √  √ ⎟  4 5 3 1 ⎟ ⎠ 15 − φ (x1 y4 + x4 y1 ) + 2(x2 y3 + x3 y2 ) − √ 3 (3φ − 1)(x4 y5 + x5 y4 )   4 15 1 √ x 1 y 5 + x 5 y1 + (−11x1 y1 + 4x2 y2 + 11x5 y5 ) − 45 (2 − 3φ)x3 y3 + (3φ − 1)x4 y4 3 15

5

√ √ ⎞ √ − 3 4 5 (x1 y1 − x5 y5 ) − 43 (x1 y5 + x5 y1 ) + 5x2 y2 − φ 2 x3 y3 + 12 x4 y4 φ √ √ ⎜ ⎟ ⎜ ⎟ 5(x1 y2 + x2 y1 ) + 3(x2 y5 + x5 y2√ ) + x3 y4 + x4 y3 ⎜ ⎟ ⎜ ⎟ 2 (x y + x y ) + x y + x y + 3 (x y + x y ) −φ ⎜ ⎟ . 1 3 3 1 2 4 4 2 3 5 5 3 φ √ ⎜ ⎟ 1 (x y + x y ) + (x y + x y ) − 3φ(x y + x y ) ⎜ ⎟ 4 1 2 3 3 2 4 5 5 4 ⎝ √ ⎠ φ2 1 4 √ √ √ − 43 (x1 y1 − 4x2 y2 − x5 y5 ) + 3 4 5 (x1 y5 + x5 y1 ) + φ3 x3 y3 − 3φx4 y4 5

⎛

⊕

(D.15)

D.2 Basis II Here we consider another basis, which is used, e.g., in [2], with the generators a and b satisfying (D.16) a 2 = b3 = (ab)5 = e. These generators a and b are given in terms of s and t of Sect. D.1 by a = st 3 st 2 s,

ab = t 4 .

(D.17)

In the above transformation, a is diagonal but ab is not. We diagonalize ab with the unitary transformation Uφ : a = Uφ† st 3 st 2 sUφ ,

ab = Uφ† t 4 Uφ .

These generators are represented as follows: ⎛ √ √ ⎞ ⎛ 1 − 2 − 2 1 0 √ 1 ⎜ 1 ⎟ a = √ ⎝ −√2 −φ ab = ⎝ 0 ρ φ ⎠, 5 − 2 1 0 0 −φ φ with

⎛

−



2 φ

√ − φ

1 ⎜ Uφ = √ ⎝ 0 i51/4 √ 251/4 √1 − 2φ φ ⎛ √ ⎞ √ 2 2 −1 1 ⎜√ ⎟ 1 2 − φ a = √ ⎝√ ⎠, φ 5 2 φ − φ1

(D.18)

⎞ 0 0 ⎠, ρ4

on 3,

√ ⎞ − φ ⎟ −i51/4 ⎠ ,

(D.19)

(D.20)

√1 φ

⎛

1 0 ab = ⎝ 0 ρ 2 0 0

⎞ 0 0 ⎠, ρ3

on 3 , (D.21)

254

D Representations of A5 in Different Bases

with ⎛ 1 Uφ = √ 251/4 ⎛

0 √ ⎜ 2φ ⎝ 2 φ

i51/4 − √1φ √ φ

⎞ −i51/4 1 − √φ ⎟ ⎠, √ φ

⎞ φ −1 1 −1 1 φ ⎟ 1 ⎜ ⎜ ⎟ a=√ ⎜ φ ⎟, 1 −1 φ1 ⎠ 5⎝ φ 1 −1 φ 1 φ

⎛

1 φ

1

(D.22)

ρ ⎜0 ab = ⎜ ⎝0 0

0 ρ2 0 0

0 0 ρ3 0

⎞ 0 0⎟ ⎟, 0⎠ ρ4

on 4,

(D.23) with

⎛

1

1⎜ ⎜ 51/4 φ 3/2 ⎜ 3/2 2 ⎝ iφ1/4 i

Uφ =

5

1

−1

−1

iφ 3/2 51/4 − 51/4iφ 3/2

3/2

51/4 φ 3/2

1

1

− iφ51/4 i

1

⎞

− 51/4iφ 3/2 ⎟ ⎟ ⎟, iφ 3/2 ⎠ − 51/4 1

(D.24)

and √ √ √ √ ⎞ −1 6 6 6 6 √ 1 2 2 ⎟ ⎜ 6 −2φ φ ⎜√ ⎟ φ φ2 1⎜ 1 2 ⎟ 2 ⎜ ⎟ 6 −2φ φ a = ⎜√ φ ⎟, φ2 5⎜ 1 2 2 φ −2φ ⎟ ⎝ √6 ⎠ φ φ2 2 1 6 φ2 −2φ 2 φ φ ⎛ ⎞ 1 0 0 0 0 ⎜0 ρ 0 ⎟ 0 0 ⎜ ⎟ 2 ⎟ , on 5, 0 0 0 0 ρ ab = ⎜ ⎜ ⎟ ⎝ 0 0 0 ρ3 0 ⎠ 0 0 0 0 ρ4 ⎛

(D.25)

with ⎛

− φ1



⎜ ⎜ 0 1 ⎜ ⎜ Uφ = √ ⎜ 0 10 ⎜ ⎜ √ 6 ⎝ φ2 √ 2

where φ = (1 +

√

3 2

1 1/4 i5 √ φ √ i51/4 φ

  − 12 φ 4 + 8 − 12 φ 4 + 8 √ √ −i51/4 φ i51/4 φ

−1 √ 3

5)/2 and ρ = e2iπ/5 .

1/4 i5 √ φ

−1 √ 3 2φ

1/4

− i5√φ −1 √ 3 2φ

1

⎞

⎟ 1/4 − i5√φ ⎟ ⎟ √ ⎟ , −i51/4 φ ⎟ ⎟ ⎟ −1 ⎠ √ 3 (D.26)

D.2 Basis II

255

The tensor products are as follows: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ y1 x2 y3 − x3 y2 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = (x1 y1 + x2 y3 + x3 y2 )1 ⊕ ⎝ x1 y2 − x2 y1 ⎠ x3 3 y3 3 x3 y1 − x1 y3 3 ⎛

⎞ 2x√ − x3 y2 1 y1 − x2 y3√ ⎜ − 3x1 y2 − 3x2 y1 ⎟ ⎜ ⎟ √ ⎟ , ⊕⎜ ⎜ ⎟ √6x2 y2 ⎝ ⎠ 6x3 y√ 3 √ − 3x1 y3 − 3x3 y1 5

(D.27)

⎛

⎛ ⎞ ⎛ ⎞ ⎞ x1 y1 x2 y3 − x3 y2 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = (x1 y1 + x2 y3 + x3 y2 )1 ⊕ ⎝ x1 y2 − x2 y1 ⎠ x3 3 y3 3 x3 y1 − x1 y3 3 ⎛

⎞ 2x1 y1 − √ x2 y3 − x3 y2 ⎜ ⎟ 6x3 y√ 3 ⎜ √ ⎟ ⎜ ⊕ ⎜ −√3x1 y2 − √3x2 y1 ⎟ ⎟ , ⎝ − 3x1 y3 − 3x3 y1 ⎠ √ 6x2 y2 5

(D.28)

√ ⎞ ⎛ ⎞ ⎛ √ 3x√ 1 y1 √2x2 y1 + x3 y2 ⎜ x2 y1 − 2x3 y2 ⎟ x1 y1 ⎟ ⎜ ⎜ − 2x1 y2 − x3 y3 ⎟ √ ⎟ ⊕ ⎜ x1 y2 − 2x3 y3 ⎟ , (D.29) ⎝ x2 ⎠ ⊗ ⎝ y 2 ⎠ = ⎜ √ ⎟ ⎜ ⎝ − 2x1 y3 − x2 y2 ⎠ √ ⎝ x1 y3 − 2x2 y2 ⎠ √ x3 3 y3 3 √ 2x3 y1 + x2 y3 4 x3 y1 − 2x2 y3 5 ⎛

⎞

⎛

⎞

√ ⎛ ⎞ ⎛ ⎞ √ ⎛ √ ⎞ ⎞ x1 y1 − √2x3 y2 y1 − 2x y − 2x y x1 2 4 3 1 ⎜ ⎟ ⎜ ⎟ √ ⎝ x2 ⎠ ⊗ ⎜ y2 ⎟ = ⎝ 2x1 y2 − x2 y1 + x3 y3 ⎠ ⊕ ⎜ −x1 y2 −√ 2x2 y1 ⎟ ⎝ y3 ⎠ ⎝ √ x1 y3 + √2x3 y4 ⎠ x3 3 2x y + x y − x y 1 3 2 2 3 4  3 y4 4 −x1 y4 + 2x2 y3 ⎛

4

√ √ ⎞ 6x2 y4 − 6x3 y1 √ ⎜ ⎟ ⎜ √2 2x1 y1 + 2x3 y2 ⎟ ⎜ ⊕ ⎜ −√ 2x1 y2 + x2 y1 + 3x3 y3 ⎟ ⎟ , ⎝ 2x1 y3 − 3x2 y2 − x3 y4 ⎠ √ −2 2x1 y4 − 2x2 y3 5 ⎛

(D.30)

256

D Representations of A5 in Different Bases

√ ⎞ ⎞ ⎛ √ ⎛ √ ⎞ x1 y1 + √2x3 y3 y1 − 2x y − 2x y x1 2 3 3 2 ⎟ ⎜ ⎟ ⎜ √ ⎝ x2 ⎠ ⊗ ⎜ y2 ⎟ = ⎝ 2x1 y1 + x2 y4 − x3 y3 ⎠ ⊕ ⎜ x1 y2 − √2x3 y4 ⎟ ⎝ y3 ⎠ ⎝ −x1 y3 + 2x2 y1 ⎠ √ √ x3 3 2x1 y4 − x2 y2 + x3 y1 3 y4 4 −x1 y4 − 2x2 y2 4 √ √ ⎛ ⎞ 6x y − 6x3 y2 2 3 √ ⎜ 2x1 y1 − 3x2 y4 − x3 y3 ⎟ ⎜ ⎟ √ ⎟ , ⊕⎜ (D.31) 2 √2x1 y2 + 2x3 y4 ⎜ ⎟ ⎝ −2 2x1 y3 − 2x2 y1 ⎠ √ − 2x1 y4 + x2 y2 + 3x3 y1 5 ⎛ ⎞ y1 √ √ ⎛ ⎛ ⎞ ⎞ ⎜ y2 ⎟ −2x x1 ⎜ ⎟ √ 1 y1 + 3x2 y5 +√ 3x3 y2 ⎝ x2 ⎠ ⊗ ⎜ y 3 ⎟ = ⎝ ⎠ ⎜ ⎟ √3x1 y2 + x √2 y1 − 6x3 y3 x3 3 ⎝ y 4 ⎠ 3x1 y5 − 6x2 y4 + x3 y1 3 y5 5 ⎛

⎞

⎛

⎞ ⎛ √ 3x1 y√1 + x2 y5 +√x3 y2 ⊕ ⎝ x1 y3 − √2x2 y2 − √2x3 y4 ⎠ x1 y4 − 2x2 y3 − 2x3 y5 3 √ √ ⎞ 2 √2x1 y2 − 6x2 y1 + x3 y3 ⎜ − 2x1 y3 + 2x2 y2 − 3x3 y4 ⎟ ⎟ √ ⊕⎜ ⎠ ⎝ 2x y + 3x y − 2x y 1 4 2 3 3 5 √ √ −2 2x1 y5 − x2 y4 + 6x3 y1 4 ⎛

√ ⎞ 3x2√y5 − 3x3√y2 ⎜ −x1 y2 − 3x2 y1 − 2x3 y3 ⎟ ⎜ ⎟ √ ⎟ , ⊕⎜ −2x1 y3 −√ 2x2 y2 ⎜ ⎟ ⎝ ⎠ 2x1√ y4 + 2x3√ y5 x1 y5 + 2x2 y4 + 3x3 y1 5 ⎛

⎛

√

⎞ y1 ⎛ √ ⎞ ⎜ y2 ⎟ 3x1 y√1 + x2 y4 +√x3 y3 x1 ⎜ ⎟ ⎝ x2 ⎠ ⊗ ⎜ y3 ⎟ = ⎝ x1 y2 − 2x2 y5 − 2x3 y4 ⎠ ⎜ ⎟ √ √ x3 3 ⎝ y4 ⎠ x1 y5 − 2x2 y3 − 2x3 y2 3 y5 5 ⎛

⎞

√ √ ⎞ −2x y + 3x y + 1 1 2 4 √ √ 3x3 y3 ⎠ ⊕ ⎝ √3x1 y3 + x √2 y1 − 6x3 y5 3x1 y4 − 6x2 y2 + x3 y1 3 ⎛

(D.32)

D.2 Basis II

257

⎛ √ ⎞ √2x1 y2 + 3x √ 2 y5 − 2x3 y4 ⎜ 2 2x1 y3 − 6x2 y1 + x3 y5 ⎟ ⎟ √ √ ⊕⎜ ⎝ −2 2x1 y4 − x2 y2 + 6x3 y1 ⎠ √ − 2x1 y5 + 2x2 y3 − 3x3 y2 4 √ √ ⎞ 3x2 y4 −√ 3x3 y3 ⎟ ⎜ 2x1 y√2 + 2x3 y√4 ⎟ ⎜ ⎜ ⊕ ⎜ −x1 y3 −√ 3x2 y1 −√ 2x3 y5 ⎟ ⎟ , ⎝ x1 y4 + 2x2 y2 + 3x3 y1 ⎠ √ −2x1 y5 − 2x2 y3 5 ⎛

⎛

(D.33)

⎞ ⎛ ⎞ x1 y1 ⎜ x2 ⎟ ⎜ y2 ⎟ ⎜ ⎟ ⊗ ⎜ ⎟ = (x1 y4 + x2 y3 + x3 y2 + x4 y1 )1 ⎝ x3 ⎠ ⎝ y3 ⎠ x4 4 y4 4

⎞ −x1 y√ x3 y2 + x4 y1 4 + x2 y3 −√ ⎠ ⊕⎝ √2x2 y4 − √2x4 y2 2x1 y3 − 2x3 y1 3 ⎛

⎛

⎞ x1 y4√+ x2 y3 − √ x3 y2 − x4 y1 ⎠ ⊕⎝ √2x3 y4 − √2x4 y3 2x1 y2 − 2x2 y1 3 ⎛

⎞ x2 y4 + x3 y3 + x4 y2 ⎜ x1 y1 + x3 y4 + x4 y3 ⎟ ⎟ ⊕⎜ ⎝ x1 y2 + x2 y1 + x4 y4 ⎠ x1 y3 + x2 y2 + x3 y1 4 √ √ √ ⎛√ ⎞ 3x1 y√4 − 3x2 y3√− 3x3 y2√+ 3x4 y1 ⎜ ⎟ − √ 2x2 y4 + 2√2x3 y3 − √2x4 y2 ⎜ ⎟ ⎟ , (D.34) ⊕⎜ −2 2x y + 2x y + 2x y 1 1 √ 3 4 √ 4 3 ⎜ ⎟ √ ⎝ ⎠ 2x y + 2x y − 2 2x y 1 2 2 1 4 4 √ √ √ − 2x1 y3 + 2 2x2 y2 − 2x3 y1 5

⎛ ⎞ ⎞ y1 √ √ √ ⎞ ⎛ √ x1 ⎜ y2 ⎟ 2 2x1√y5 − 2x2 y4 + 2x3 y3 − 2 2x4 y2 ⎜ ⎟ ⎜ x2 ⎟ ⎜ ⎟ ⊗ ⎜ y3 ⎟ = ⎝ ⎠ − 6x1 y1 + 2x2 y5 + 3x3 y√ 4 − x4 y3 ⎜ ⎟ ⎝ x3 ⎠ ⎝ y4 ⎠ x1 y4 − 3x2 y3 − 2x3 y2 + 6x4 y1 3 x4 4 y5 5 ⎛

258

D Representations of A5 in Different Bases

⎛√

√ √ √ ⎞ 2x1 y5 + 2 √ 2x2 y4 − 2 2x3 y3 − 2x4 y2 ⎠ ⊕⎝ 3x1 y2 − 6x2 y1 − √ x3 y5 + 2x4 y4 −2x1 y3 + x2 y2 + 6x3 y1 − 3x4 y5 3 √ √ √ ⎞ ⎛ √ 3x y − 2x y + 2x y − 2 1 1 2 5 3 4 √ √ √2x4 y3 √ ⎜ − 2x1 y2 − 3x2 y1 + 2 2x3 y5 + 2x4 y4 ⎟ ⎟ √ √ √ √ ⊕⎜ ⎝ 2x1 y3 + 2 2x2 y2 − 3x3 y1 − 2x4 y5 ⎠ √ √ √ √ −2 2x1 y4 + 2x2 y3 − 2x3 y2 + 3x4 y1 4 ⎛√

√ √ √ ⎞ 2x1 y√ 5 − 2x2 y√ 4 − 2x3 y√ 3 + 2x4 y2 ⎟ ⎜ −√ 2x1 y1 −√ 3x3 y4 −√ 3x4 y3 ⎟ ⎜ ⎟ ⎜ ⊕⎜ 3x1 y2 + √2x2 y1 + √3x3 y5 ⎟ √ ⎠ ⎝ √3x2 y2 + √2x3 y1 + √3x4 y5 − 3x1 y4 − 3x2 y3 − 2x4 y1 5 ⎛ ⎞ 2x1 y5 + 4x2 y4 +√ 4x3 y3 + 2x4 y2 ⎜ ⎟ 4x y + 2 1 1 ⎜ √ ⎟ √ 6x2 y5 √ ⎜ ⎟ ⊕ ⎜ − √6x1 y2 + 2x √ 2 y1 − 6x3 y5 + 2√ 6x4 y4 ⎟ , ⎝ 2 6x1 y3 − 6x2 y2 + 2x3 y1 − 6x4 y5 ⎠ √ 2 6x3 y2 + 4x4 y1 5 ⎛

⎞

⎛

⎞

x1 y1 ⎜ x2 ⎟ ⎜ y2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x3 ⎟ ⊗ ⎜ y 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ x4 ⎠ ⎝ y4 ⎠ x5 5 y5 5 = (x1 y1 + x2 y5 + x3 y4 + x4 y3 + x5 y2 )1 ⎛ ⎞ 2 √4 y3 − x5 y√ √ x2 y5 +√2x3 y4 − 2x ⊕ ⎝ −√ 3x1 y2 +√ 3x2 y1 +√ 2x3 y5 −√ 2x5 y3 ⎠ 3x1 y5 + 2x2 y4 − 2x4 y2 − 3x5 y1 3 ⎞ ⎛ y2 √4 y3 − 2x5√ √ 2x2 y5√− x3 y4 + x ⊕ ⎝ √3x1 y3 − √3x3 y1 + √2x4 y5 − √2x5 y4 ⎠ − 3x1 y4 + 2x2 y3 − 2x3 y2 + 3x4 y1 3 √ √ √ √ ⎞ ⎛ √ 3x3 y5 + 4√3x4 y4 − √3x5 y3 3√2x1 y2 + 3√2x2 y1 − √ ⎜ 3 2x1 y3 + 4 3x2 y2 + 3 2x3 y1 − 3x4 y5 − 3x5 y4 ⎟ ⎟ √ √ √ √ √ ⊕⎜ ⎝ 3 2x1 y4 − 3x2 y3 − 3x3 y2 + 3 2x4 y1 + 4 3x5 y5 ⎠ √ √ √ √ √ 3 2x1 y5 − 3x2 y4 + 4 3x3 y3 − 3x4 y2 + 3 2x5 y1 4

(D.35)

References

√ √ √ ⎞ ⎛ √ √2x1 y2 − √2x2 y1 + √3x3 y5 − √3x5 y3 ⎜ − 2x1 y3 + 2x3 y1 + 3x4 y5 − 3x5 y4 ⎟ ⎟ √ √ √ √ ⊕⎜ ⎝ − 2x1 y4 − 3x2 y3 + 3x3 y2 + 2x4 y1 ⎠ √ √ √ √ 2x1 y5 − 3x2 y4 + 3x4 y2 − 2x5 y1 4 ⎞ ⎛ 2x1 y1 + x2 y5 − 2x y3 + x5 y2 √3 y4 − 2x4√ ⎜ x1 y2 + x2 y1 + 6x3 y5 + 6x5 y3 ⎟ ⎟ ⎜ √ ⎟ −2x1 y3 + 6x2 y2 − ⊕⎜ ⎟ ⎜ √2x3 y1 ⎠ ⎝ −2x√ 1 y4 − 2x4 y√ 1 + 6x5 y5 x1 y5 + 6x2 y4 + 6x4 y2 + x5 y1 5 ⎛ ⎞ 2x1 y1 − 2x2 y5 + x3 y4 + √ x4 y3 − 2x5 y2 ⎜ ⎟ −2x y − 2x y + 6x 1 2 ⎜ ⎟ √2 1 √4 y4 ⎟ . x y + x y + 6x y + 6x y ⊕⎜ 1 3 3 1 4 5 5 4 ⎜ ⎟ √ √ ⎝ x1 y4 + 6x2 y3 + 6x3 y2 + x4 y1 ⎠ √ −2x1 y5 + 6x3 y3 − 2x5 y1 5

References 1. Shirai, K.: J. Phys. Soc. Jpn. 61, 2735 (1992) 2. Ding, G.-J., Everett, L.L., Stuart, A.J.: arXiv:1110.1688 [hep-ph]

259

(D.36)

Appendix E

Representations of T  in Different Bases

Here we consider another basis for representations of the T  group. All elements of T  can be written as products of the generators s and t , which satisfy s 2 = r,

r 2 = t 3 = (st)3 = e,

rt = tr.

(E.1)

In Chap. 5, the doublet and triplet representations were as follows:  t=  t=  t=

ω2 0

1 0

ω 0

0 , ω

 r=





0 , ω2

r=





0 , 1

⎛

r= ⎞

1 0 0 ⎝ t = 0 ω 0 ⎠, 0 0 ω2 ⎛ −1 2p1 1 −1 s = ⎝ 2p¯ 1 3 2p¯ p¯ 2p¯ 1 2

2

−1 0 , 0 −1

1 s = −√ 3



−1 0 , 0 −1

1 s = −√ 3



−1 0

1 s = −√ 3

0 , −1 ⎛







√i − 2p¯

√i − 2p¯

√i − 2p¯

⎞

1 0 0 ⎝ r = 0 1 0⎠, 0 0 1 ⎞ 2p1 p2 2p2 ⎠ on 3, −1

√

2p −i

√

2p −i

√

2p −i







on 2, (E.2) on 2 , (E.3) on 2 , (E.4)

(E.5)

where p1 = eiφ1 and p2 = eiφ2 . H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5, © Springer-Verlag Berlin Heidelberg 2012

261

E Representations of T  in Different Bases

262

E.1 Basis I In this section, we recall the basis used in Chap. 5. We take the parameters p = i and p1 = p2 = 1, whereupon the generator s takes the form √  i 1 2 √ s = −√ , on 2, 2 , and 2 , (E.6) 2 −1 3 ⎛ ⎞ −1 2 2 1 s = ⎝ 2 −1 2 ⎠ , on 3. (E.7) 3 2 2 −1 The tensor products decompose as follows: 

x1 x2





y1 y2





x1 y2 − x2 y1 √ 2

⎛ x1 y2 +x2 y1 ⎞



√

2

⊕ ⎝ −x1 y1 ⎠ , 2(2 ) 2(2 ) 1 x2 y2 3 ⎛ ⎞    −x1 y1 x1 y2 − x2 y1 y x1 ⊗ 1 = ⊕ ⎝ x2 y2 ⎠ , √ x2 2 (2) y2 2 (2 ) x1 y2√ +x2 y1 2 1 

x1 x2

⊗





y ⊗ 1 y 2 2 (2)

=



 2 (2 )

=

x1 y2 − x2 y1 √ 2



2

⎛

⎞

(E.8)

(E.9)

3

x2 y 2 x1 y2√ +x2 y1 ⎠ ⎝ ⊕ , 2 1 −x1 y1 3

(E.10)

⎛ ⎞ ⎞ y1 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = [x1 y1 + x2 y3 + x3 y2 ]1 x3 3 y3 3 ⎛

⊕ [x3 y3 + x1 y2 + x2 y1 ]1 ⊕ [x2 y2 + x1 y3 + x3 y1 ]1 ⎞ ⎛ 2x y − x2 y3 − x3 y2 1⎝ 1 1 2x3 y3 − x1 y2 − x2 y1 ⎠ ⊕ 3 2x y − x y − x y ⎛

⊕ ⎛

2 2

1 3

3 1

⎞

x y − x3 y2 1⎝ 2 3 x1 y2 − x2 y1 ⎠ , 2 x y −x y 3 1

1 3

3

3

(E.11)

⎞ √ √ y1 2x y + x1 y1 2x y + x1 y2 x1 ⊗ ⎝ y2 ⎠ = √ 2 2 ⊕ √ 2 3 x2 2,2 ,2 2x1 y3 − x2 y1 2,2 ,2 2x1 y1 − x2 y2 2 ,2 ,2 y3 3 √ 2x y + x y 2 1 1 3 ⊕ √ , (E.12) 2x1 y2 − x2 y3 2 ,2,2   y xy1 (x)1 (1 ) ⊗ 1 = , (E.13) y2 2,2 ,2 xy2 2 (2 ),2 (2),2(2 ) 



E.2

Basis II

263

⎛

⎛ ⎞ ⎞ y1 xy3 (x)1 ⊗ ⎝ y2 ⎠ = ⎝ xy1 ⎠ , y3 3 xy2 3

⎛

⎛ ⎞ ⎞ y1 xy2 (x)1 ⊗ ⎝ y2 ⎠ = ⎝ xy3 ⎠ . y3 3 xy1 3

(E.14)

E.2 Basis II We now consider another basis, which is used, e.g., in [1]. We take the parameters p = eiπ/12 and p1 = p2 = ω, whence the generator s takes the form √ iπ/12  1 2e √ i−iπ/12 (E.15) s = −√ , on 2, 2 , 2 , −i 3 − 2e ⎞ ⎛ −1 2ω 2ω2 1⎝ 2 (E.16) s= 2ω −1 2ω ⎠ , on 3. 3 2ω 2ω2 −1 The tensor products are as follows: ⎞ (x1 y2 + x2 y1 ) ⎠ , = [x1 y2 − x2 y1 ]1 ⊕ ⎝ ix1 y1 2(2 ) x2 y2 3 ⎞ ⎛   ix1 y1 x1 y ⎠ , x2 y2 ⊗ 1 = [x1 y2 − x2 y1 ]1 ⊕ ⎝ x2 2 (2) y2 2 (2 ) 1−i 2 (x1 y2 + x2 y1 ) 3 ⎞ ⎛   x2 y 2 x1 y1 ⎠ , ⊗ = [x1 y2 − x2 y1 ]1 ⊕ ⎝ 1−i 2 (x1 y2 + x2 y1 ) x2 2 (2) y2 2 (2 ) ix1 y1 3 ⎛ ⎞ ⎛ ⎞ y1 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = [x1 y1 + x2 y3 + x3 y2 ]1 x3 3 y3 3 

x1 x2





y ⊗ 1 y 2 2(2 )

⎛ 1−i



2

(E.17)

(E.18)

(E.19)

⊕ [x3 y3 + x1 y2 + x2 y1 ]1 ⊕ [x2 y2 + x1 y3 + x3 y1 ]1 ⎞ ⎛ 2x1 y1 − x2 y3 − x3 y2 1 ⊕ ⎝ 2x3 y3 − x1 y2 − x2 y1 ⎠ 3 2x y − x y − x y ⎛

⊕ 

x1 x2



⎛

⎞

2 2

1 3

⎞

3 1

x y − x3 y2 1⎝ 2 3 x1 y2 − x2 y1 ⎠ , 2 x y −x y 3 1

1 3

3

 y1 (1 + i)x2 y2 + x1 y1 ⊗ ⎝ y2 ⎠ = (1 − i)x1 y3 − x2 y1 2,2 ,2 2,2 ,2 y3 3

3

(E.20)

264

E

 ⊕ 

Representations of T  in Different Bases

(1 + i)x2 y3 + x1 y2 (1 − i)x1 y1 − x2 y2



2 ,2 ,2

(1 + i)x2 y1 + x1 y3 , (E.21) (1 − i)x1 y2 − x2 y3 2 ,2,2   y xy1 (x)1 (1 ) ⊗ 1 = , (E.22) y2 2,2 ,2 xy2 2 (2 ),2 (2),2(2 ) ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ⎛ y1 xy3 y1 xy2 (x)1 ⊗ ⎝ y2 ⎠ = ⎝ xy1 ⎠ , (x)1 ⊗ ⎝ y2 ⎠ = ⎝ xy3 ⎠ . (E.23) y3 3 xy2 3 y3 3 xy1 3 ⊕

References 1. Feruglio, F., Hagedorn, C., Lin, Y., Merlo, L.: Nucl. Phys. B 775, 120 (2007). arXiv:hep-ph/ 0702194

Appendix F

Other Smaller Groups

In this appendix, we study finite groups whose orders are less than 31 [1, 2]. Such groups are summarized in Table F.1, where g denotes the order. Here we have omitted the Abelian groups ZN as well as their direct products. Most of the finite groups in Table F.1 are non-Abelian groups mentioned in the text, and their extensions by direct products with Abelian groups such as Z2 × D4 , Z2 × Q4 , etc. However, the table includes non-Abelian groups which are not mentioned in the text, in particular, Z4 Z4 , Z8 Z2 , (Z4 ×Z2 )Z2 (I ), (Z4 ×Z2 )Z2 (I I ), Z3 Z8 , (Z6 ×Z2 )Z2 , and Z9  Z3 . In this appendix, we shall explicitly discuss these groups.

F.1 Z4  Z4 We denote the first and second Z4 generators by a and b, respectively. They thus satisfy a 4 = b4 = e. Then we have the relation ab = ba m , which implies ab2 = 2 b2 a m . We require m = 0 and m2 = 0 mod 4. If m = 1 mod 4, the generators a and b commute and the group becomes the direct product Z4 × Z4 . We thus require m = 1 mod 4. These requirements are then satisfied for m = 3 mod 4, i.e., ab = ba 3 .

(F.1)

Using these generators, all elements of Z4  Z4 can be written in the form g = bm a n ,

(F.2)

with n, m = 0, 1, 2, 3. H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5, © Springer-Verlag Berlin Heidelberg 2012

265

266

F Other Smaller Groups

Table F.1 Classification of the non-Abelian groups with g ≤ 30. Note that there are two finite groups isomorphic to (Z4 × Z2 )  Z2 apart from D8 g 6 8

Groups S3 ≡ D3 D4 , Q4

10

D5

12

A4 , D6 , Q6

14

D7

16

D8 , Q8 , QD16 , Z2 × D4 , Z2 × Q4 , Z4  Z4 , Z8  Z2 , (Z4 × Z2 )  Z2 (I ), (Z4 × Z2 )  Z2 (I I )

18

D9 , Z3 × D3 , Σ(18) ≡ (Z3 × Z3 )  Z2

20

D10 , Q10

21

T7 ≡ Z7  Z3

22

D11

24

D12 , S4 , Q12 , T   SL(2, 3), Z2 × Z2 × S3 , Z4 × S3 , Z2 × Q6 , Z3 × D4 , Z3 × Q4 , Z2 × A4 , Z3  Z8 , (Z6 × Z2 )  Z2

26

D13

27

Δ(27) ≡ (Z3 × Z3 )  Z3 , Z9  Z3

28

Q14 , D14

30

Z5 × S3 , Z3 × D5 , D15

All the elements bm a n are classified into ten conjugacy classes: C1 : (1) C1 : (2) C1 : (3) C1 : (1) C2 : C2(2) : (3) C2 : (4) C2 :  (1) C2 :  (2) C2 :

{e},  2 a ,  2 b ,  2 2 b a ,   b, ba 2 ,   ba, ba 3 ,  3 3 2 b ,b a ,   3 b a, b3 a 3 ,   a, a 3 ,   2 b a, b2 a 3 ,

h = 1, h = 2, h = 2, h = 2, h = 4, h = 4,

(F.3)

h = 4, h = 4, h = 4, h = 4.

The group Z4  Z4 has eight singlets 1±,k with k = 0, 1, 2, 3, and two doublets 21 and 22 . The generators b and a are represented by b = ik ,

a = ±1,

on 1±,k ,

(F.4)

F.1 Z4  Z4

267

Table F.2 Characters of Z4  Z4 h

χ1+,0

χ1+,1

χ1+,2

χ1+,3

χ1−,0

χ1−,1

χ1−,2

χ1−,3

χ 21

χ 22

C1

1

1

1

1

1

1

1

1

1

2

2

C1(1)

2

1

1

1

1

1

1

1

1

−2

−2

C1(2)

2

1

1

1

1

−1

−1

−1

−1

2

−2

C1

(3)

2

1

1

1

1

−1

−1

−1

−1

−2

2

C2(1)

4

1

−1

−1

1

i

−i

i

−i

0

0

C2

(2)

4

1

1

−1

−1

−i

i

i

−i

0

0

C2(3) (4) C2  C2(1)  (2) C2

4

1

−1

−1

1

−i

i

−i

i

0

0

4

1

1

−1

−1

i

−i

−i

i

0

0

4

1

−1

1

−1

−1

−1

1

1

0

0

4

1

−1

1

−1

1

1

−1

−1

0

0

and 

0 1 b= , 1 0  0 1 b= , −1 0



i 0 a= , on 21 , 0 −i  −i 0 a= , on 22 . 0 i

(F.5) (F.6)

The characters are shown in Table F.2. The tensor products between doublets are:     y x3 y x1 ⊗ 1 = ⊗ 3 x3 2 y3 2 x1 2 y1 2 1

1

2

2

= (x1 y3 + x3 y1 )1+,0 ⊕ (x1 y3 − x3 y1 )1+,2 

x1 x3



⊗ 21

y3 y1

22

⊕ (x1 y1 + x3 y3 )1+,3 ⊕ (x1 y1 − x3 y3 )1+.1 ,

(F.7)

= (x1 y3 + x3 y1 )1−,2 ⊕ (x1 y3 − x3 y1 )1−,0 ⊕ (x1 y1 + x3 y3 )1−,3 ⊕ (x1 y1 − x3 y3 )1−,1 .

The tensor products between singlets and doublets are:   y xy1(3) (x)1±,0 ⊗ 1(3) = , y3(1) 2 (2 ) xy3(1) 2 (2 ) 1 2 1 2   y xy1 (x)1±,2 ⊗ 1 = , y3 2 xy3 2 1 2   y xy3 = , (x)1±,2 ⊗ 3 y1 2 xy1 2 2

1

(F.8)

(F.9) (F.10) (F.11)

268

F Other Smaller Groups

 (x)1±,1 ⊗  (x)1±,3 ⊗

y1(3) y3(1) y1(3) y3(1)



21 (22 )

21 (22 )

= (xy1 )1±,2 ⊕ (xy3 )1±,0 ,

(F.12)

= (xy1 )1±,0 ⊕ (xy3 )1±,2 .

(F.13)

The tensor products between singlets are: 1±,i ⊗ 1±,j = 1±,i+j (mod 4) ,

1±,i ⊗ 1∓,j = 1∓,i+j (mod 4) ,

(F.14)

where i, j = 0, 1, 2, 3.

F.2 Z8  Z2 Here we study the group Z8  Z2 other than D8 and QD16 . We denote the generators of Z8 and Z2 by a and b, respectively. That is, they satisfy a 8 = e,

b2 = e.

(F.15)

In addition, we require bab = a m ,

(F.16)

m2

where m = 0. This leads to b2 ab2 = a . Since b2 = e, consistency requires m2 = 1 mod 8. Then the possible values are found to be m = 1, 3, 5, 7. However, the groups with m = 7 and 3 correspond to D8 and QD16 , respectively, while the group with m = 1 is just the direct product Z8 × Z2 . Therefore, we shall focus here on the group with m = 5. All elements of the group can be written in the form bk a  with k = 0, 1, and  = 0, . . . , 7. These elements are classified into ten conjugacy classes: C1 : (1) C2 : (2) C2 : (1) C1 : (2) C1 : (3) C1 : (1) C2 : (2) C2 : (3) C2 : (4) C2 :

{e},   a, a 5 ,  3 7 a ,a ,  2 a ,  4 a ,  6 a ,   b, ba 4 ,   ba, ba 5 ,   2 ba , ba 6 ,   3 ba , ba 7 ,

h = 1, h = 8, h = 8, h = 4, h = 2, h = 4, h = 2, h = 2, h = 2, h = 2.

(F.17)

F.2 Z8  Z2

269

Table F.3 Characters of Z8  Z2 h

χ1+0

χ1−0

χ1+1

χ1−1

χ1+2

χ1−2

χ1+3

χ1−3

χ 21

χ 22

C1

1

1

1

1

1

1

1

1

1

2

2

C2(1)

8

1

1

i

i

−1

−1

−i

−i

0

0

C2(2) (1) C1 C1(2) (3) C1 C2(1) (2) C2 C2(3) (4) C2

8

1

1

−i

−i

−1

−1

i

i

0

0

4

1

1

−1

−1

1

1

−1

−1

2i

−2i

2

1

1

1

1

1

1

1

1

−2

−2

4

1

1

−1

−1

1

1

i

i

−2i

2i

2

1

−1

1

−1

1

−1

1

−1

0

0

2

1

−1

i

−i

−1

1

−i

i

0

0

2

1

−1

−1

1

1

−1

−1

1

0

0

2

1

−1

−i

i

−1

1

i

−i

0

0

The group Z8  Z2 has eight singlets 1±,k with k = 0, 1, 2, 3, and two doublets 21 and 22 . The characters are shown in Table F.3. The generators a and b can be represented by a = ik ,

b = ±1,

on 1±,k .

(F.18)

They can also be represented by the following matrices: 0 , ρ5  7 0 ρ , a= 0 ρ3 

a=

ρ 0



0 1 , on 21 , 1 0  0 1 b= , on 22 . 1 0

b=

(F.19) (F.20)

The tensor products between doublets are:   y x1 ⊗ 1 = (x1 y1 + x2 y2 )1+,1 ⊕ (x1 y1 − x2 y2 )1−,1 x2 2 y2 2 1





x1 x2

x1 x2

1



 ⊗

21



 ⊗

22

y1 y2

y1 y2

⊕ (x1 y2 + x2 y1 )1+,3 ⊕ (x1 y2 − x2 y1 )1−,3 ,

22

= (x1 y1 + x2 y2 )1+,0 ⊕ (x1 y1 − x2 y2 )1−,0 ⊕ (x1 y2 + x2 y1 )1+,2 ⊕ (x1 y2 − x2 y1 )1−,2 ,

22

(F.21)

(F.22)

= (x1 y1 + x2 y2 )1+,3 ⊕ (x1 y1 − x2 y2 )1−,3 ⊕ (x1 y2 + x2 y1 )1+,1 ⊕ (x1 y2 − x2 y1 )1−,1 .

The tensor products between doublets and singlets are:

(F.23)

270

F Other Smaller Groups

 (x)1±,0 ⊗  (x)1±,1 ⊗  (x)1±,2 ⊗  (x)1±,3 ⊗

y1 y2 y1 y2

y1 y2

y1 y2



 =



2i

 =

21



 =

21



 =

21

xy1 ±xy2 xy2 ±xy1

xy2 ±xy1

xy1 ±xy2

(F.24)

,

2i

 (x)1±,1 ⊗

, 22



 (x)1±,2 ⊗

, 21



 (x)1±,3 ⊗

, 22

y1 y2

y1 y2

y1 y2



 = 22



 = 22



 = 22

xy1 ±xy2

xy2 ±xy1

xy2 ±xy1

, 21

(F.25)



, 22

(F.26)



. 21

(F.27) The tensor products between singlets are 1s,k ⊗ 1s  ,k  = 1s  ,k+k  ,

(F.28)

where s  = ss  .

F.3 (Z2 × Z4 )  Z2 (I) Here we discuss the group (Z2 × Z4 )  Z2 (I). We denote the first and second Z2 generators by a and b, respectively, while the generator of Z4 is written a. ˜ The generators a, a, ˜ and b satisfy the conditions a 2 = e,

a˜ 4 = e,

b2 = e,

ab ˜ = ba. ˜ (F.29) All elements can written bk a  a˜ m with k = 0, 1,  = 0, 1, and m = 0, 1, 2, 3. These elements are classified into ten conjugacy classes: C1 : C1(1) (2) C1 (3) C1 (1) C2 (2) C2 C2(3) C2(4) (5) C2 (6) C2

{e},

bab = a a˜ 2 ,

a a˜ = aa, ˜

h = 1,

: {a}, ˜ h = 4,  2 h = 2, : a˜ ,  3 h = 4, : a˜ ,   h = 2, : a, a a˜ 2 ,   h = 4, : a a, ˜ a a˜ 3 ,   2 : b, a˜ b , h = 2,   2 : ab, a a˜ b , h = 4,   3 : ab, ˜ a˜ b , h = 4,   3 : a ab, ˜ a a˜ b , h = 2.

(F.30)

F.3 (Z2 × Z4 )  Z2 (I)

271

Table F.4 Characters of (Z2 × Z4 )  Z2 (I) h

χ1+++

χ1++−

χ1+−+

χ1+−−

χ1−++

χ1−+−

χ1−−+

χ1−−−

χ2 1

χ 22

1

1

1

1

1

1

1

1

1

2

2

C1

(1)

4

1

1

−1

−1

1

1

−1

−1

2i

−2i

C1(2)

2

1

1

1

1

1

1

1

1

−2

−2

C1

(3)

4

1

1

−1

−1

1

1

−1

−1

−2i

2i

C2(1)

2

1

1

1

1

−1

−1

−1

−1

0

0

C2(2)

4

1

1

−1

−1

−1

−1

1

1

0

0

C2(3) C2(4) C2(5) C2(6)

2

1

−1

1

−1

1

−1

1

−1

0

0

4

1

−1

1

−1

−1

1

−1

1

0

0

4

1

−1

−1

1

1

−1

−1

1

0

0

2

1

−1

−1

1

−1

1

1

−1

0

0

C1

The group (Z2  Z4 )  Z2 (I) has eight singlets 1±±± and two doublets 21 and 22 . The characters are shown in Table F.4. Regarding the singlets, the generators a, a, ˜ and b can be represented by a = ±1,

on 1±ss  ,

(F.31)

a˜ = ±1,

on 1s±s  ,

(F.32)

b = ±1,

on 1ss  ± ,

(F.33)

for any s and s  ,

for any s and s  , and

for any s and s  . For the doublets, the generator a˜ can be represented by  i 0 a˜ = , on 21 , 0 i  a˜ =

−i 0

0 , −i

on 22 .

The generators a and b can be represented by   1 0 0 1 a= , b= , 0 −1 1 0 on both doublets. The tensor products between doublets are:   y x1 ⊗ 1 = (x1 y1 + x2 y2 )1+−+ ⊕ (x1 y1 − x2 y2 )1+−− x2 2 y2 2 1

1

(F.34)

(F.35)

(F.36)

272

F Other Smaller Groups





x1 x2

x1 x2





y1 y2

⊗ 21





y1 y2

⊗ 22

⊕ (x1 y2 + x2 y1 )1−−+ ⊕ (x1 y2 − x2 y1 )1−−− , (F.37)

22

= (x1 y1 + x2 y2 )1+++ ⊕ (x1 y1 − x2 y2 )1++− ⊕ (x1 y2 + x2 y1 )1−++ ⊕ (x1 y2 − x2 y1 )1−+− , (F.38)

22

= (x1 y1 + x2 y2 )1+−+ ⊕ (x1 y1 − x2 y2 )1+−− ⊕ (x1 y2 + x2 y1 )1−−+ ⊕ (x1 y2 − x2 y1 )1−−− . (F.39)

The tensor products between doublets and singlets are:  (x)1++± ⊗

y1 y2



y (x)1+−± ⊗ 1 y2  (x)1−+± ⊗ 

y1 y2

y (x)1−−± ⊗ 1 y2



=

2i

 =

21



=

2i

 =

21

xy1 ±xy2 xy1 ±xy2

xy2 ±xy1 xy2 ±xy1

(F.40)

,

2i



y (x)1+−± ⊗ 1 y2

, 22



= 22

xy1 ±xy2

, 21

(F.41)



(F.42)

,

2i



y (x)1−−± ⊗ 1 y2

, 22



 = 22

xy2 ±xy1

. 21

(F.43) The tensor products between singlets are 1s1 s2 s3 ⊗ 1s1 s2 s3 = 1s1 s2 s3 ,

(F.44)

where s1 = s1 s1 , s2 = s2 s2 , and s3 = s3 s3 .

F.4 (Z2 × Z4 )  Z2 (II) We now turn to (Z2 × Z4 )  Z2 (II). We denote the first and second Z2 generators ˜ The generators a, a, ˜ and b satisfy by a and b, while the generator of Z4 is written a. the conditions a 2 = e,

a˜ 4 = e,

b2 = e,

bab ˜ = a a, ˜

a a˜ = aa, ˜

ab = ba. (F.45)

F.4

(Z2 × Z4 )  Z2 (II)

273

Table F.5 Characters of (Z2 × Z4 )  Z2 (II) h

χ1+0

χ1+1

χ1+2

χ1+3

χ1−0

χ1−1

χ1−2

χ1−3

χ2 1

χ 22

1

1

1

1

1

1

1

1

1

2

2

C1

(1)

2

1

1

1

1

1

1

1

1

−2

−2

C1(2)

2

1

−1

1

−1

1

−1

1

−1

2

−2

C1

(3)

2

1

−1

1

−1

1

−1

1

−1

−2

2

C2(1)

4

1

i

−1

−1

1

i

−1

−i

0

0

C2(2)

4

1

−i

−1

i

1

−i

−1

i

0

0

C2(3)

2

1

1

1

1

−1

−1

−1

−1

0

0

C2(4)

4

1

i

−1

−i

−1

−i

1

i

0

0

C2(5)

4

1

−1

1

−1

−1

1

−1

1

0

0

C2(6)

4

1

−i

−1

i

−1

i

1

−i

0

0

C1

All elements can be written in the form bk a  a˜ m with k,  = 0, 1, and m = 0, 1, 2, 3. These elements are classified into ten conjugacy classes: C1 :

{e},

h = 1,

(1)

C1 : {a},   (2) C1 : a˜ 2 ,   C1(3) : a a˜ 2 ,

h = 2,

(1) C2 (2) C2 (3) C2 (4) C2 (5) C2 (6) C2

: {a, ˜ a a}, ˜   3 : a˜ , a a˜ 3 ,

h = 4,

: {b, ab},

h = 2,

h = 2, h = 2, h = 4,

(F.46)

: {ab, ˜ a ab}, ˜ h = 4,  2  : a˜ b, a a˜ 2 b , h = 2,   : a˜ 3 b, a a˜ 3 b , h = 2.

The group (Z2  Z4 )  Z2 (II) has eight singlets 1±,k with k = 0, 1, 2, 3, and two doublets 21 and 22 . The characters are shown in Table F.5. The generators, a, a˜ and b, can be represented by a = 1,

a˜ = ik ,

b = ±1,

on 1±,k .

(F.47)

In addition, the generator a˜ can be represented by  a˜ =

1 0 , 0 −1

on 21 ,

(F.48)

274

F Other Smaller Groups

 a˜ =

0 , −i

i 0

on 22 ,

(F.49)

and for both doublets the generators a and b can be represented by  a=

−1 0 , 0 −1

 b=

0 1 . 1 0

(F.50)

The tensor products between doublets are: 





x1 x2

x1 x2

x1 x2





21

y ⊗ 1 y2



 ⊗

21



⊗ 22

y1 y2

y1 y2

21

= (x1 y1 + x2 y2 )1+,0 ⊕ (x1 y1 − x2 y2 )1−,0 ⊕ (x1 y2 + x2 y1 )1+,2 ⊕ (x1 y2 − x2 y1 )1−,2 ,

22

(F.51)

= (x1 y1 + x2 y2 )1+,1 ⊕ (x1 y1 − x2 y2 )1−,1 ⊕ (x1 y2 + x2 y1 )1+,3 ⊕ (x1 y2 − x2 y1 )1−,3 ,

22

(F.52)

= (x1 y1 + x2 y2 )1+,2 ⊕ (x1 y1 − x2 y2 )1−,2 ⊕ (x1 y2 + x2 y1 )1+,0 ⊕ (x1 y2 − x2 y1 )1−,0 .

(F.53)

The tensor products between doublets and singlets are:  y1 xy1 (x)1±,0 ⊗ = , y2 2 ±xy2 2 i i   y1 xy1 = , (x)1±,1 ⊗ y2 2 ±xy2 2 

1

 (x)1±,2 ⊗ 

y1 y2

y (x)1±,3 ⊗ 1 y2

2



=

2i

 =

21

xy2 ±xy1 xy2 ±xy1

(F.54) 

y (x)1±,1 ⊗ 1 y2



= 22

xy2 ±xy1

, 21

(F.55)



(F.56)

,

2i



, 22

y (x)1±,3 ⊗ 1 y2



= 22

xy1 ±xy2

. 21

(F.57) The tensor products between singlets are 1s,k ⊗ 1s  ,k  = 1s  ,k+k  , where s  = ss  .

(F.58)

F.5 Z3  Z8

275

F.5 Z3  Z8 Here we denote the Z3 and Z8 generators by a and b, respectively. These generators satisfy the conditions a 3 = e,

b8 = e,

b−1 ab = a 2 .

(F.59)

All elements can be written bk a  with k = 0, . . . , 7, and  = 0, 1, 2. These elements are classified into twelve conjugacy classes: C1 : (1) C1 C1(2) C1(3) (1) C2 (2) C2 C2(3) (4) C2 (1) C3 (2) C3 C3(3) (4) C3

: : : : : : : : : : :

{e},  2 b ,  4 b ,  6 b ,   a, a 2 ,   2 b a, b2 a 2 ,   4 b a, b4 a 2 ,   6 b a, b6 a 2 ,   b, ba, ba 2 ,   3 3 b , b a, b3 a 2 ,   5 5 b , b a, b5 a 2 ,   7 7 b , b a, b7 a 2 ,

h = 1, h = 4, h = 2, h = 4, h = 3, h = 12, h = 3,

(F.60)

h = 3, h = 8, h = 8, h = 8, h = 8.

The group Z3  Z8 has eight singlets 1r with r = 0, . . . , 7, and four doublets 2k with k = 1, 2, 3, 4. The characters are shown in Table F.6. The generators a and b can be represented by a = 1,

b = ρk ,

on 1k ,

(F.61)

where ρ = eπi/4 . The generators a and b are also represented by the following matrices:   ω 0 0 1 (F.62) a= , b = , on 21 , 1 0 0 ω2   ω 0 0 1 , b = a= , on 22 , (F.63) i 0 0 ω2   ω 0 0 1 , b= a= , on 23 , (F.64) −1 0 0 ω2   ω 0 0 1 , b= (F.65) , on 24 , a= −i 0 0 ω2

276

F Other Smaller Groups

Table F.6 Characters of Z3  Z8 h

χ 10

χ1 1

χ 12

χ1 3

χ1 4

χ 15

χ1 6

χ 17

χ2 1

χ 22

χ2 3

χ 24

C1

1

1

1

1

1

1

1

1

1

2

2

2

2

C1(1)

8

1

i

−1

−i

1

i

−1

−i

2

2i

−2

−2i

C1(2)

8

1

−1

1

−1

1

−1

1

−1

2

−2

2

−2

C1

(3)

8

1

−i

−1

i

1

−i

−1

i

2

−2i

−2

2i

C2(1)

3

1

1

1

1

1

1

1

1

−1

−1

−1

−1

(2)

12

1

i

−1

−i

1

i

−1

−i

−2

−2i

2

2i

C2(3) (4) C2 C3(1) (2) C3 C3(3) (4) C3

3

1

−1

1

−1

1

−1

1

−1

−2

2

−2

2

3

1

−i

−1

i

1

−i

−1

i

−2

2i

2

−2i

C2

8

1

ρ

i

iρ

−1

−ρ

−i

−iρ

0

0

0

0

12

1

iρ

−i

ρ

−1

−iρ

i

−ρ

0

0

0

0

8

1

−ρ

i

−iρ

−1

ρ

−i

iρ

0

0

0

0

8

1

−iρ

−i

−ρ

−1

iρ

i

ρ

0

0

0

0

where ω = e2πi/3 . The tensor products between doublets are:  









x1 x2 x1 x2

x1 x2

x1 x2

x1 x2

x1 x2



 ⊗

21



 ⊗

21



 ⊗

21



 ⊗

21



 ⊗

22



 ⊗

22

y1 y2 y1 y2

y1 y2

y1 y2

y1 y2

y1 y2



 =



21

 =

22



 = 23



 = 24



 = 22



 = 23

x2 y2 x1 y1 x2 y1 x1 y2

⊕ (x1 y2 + x2 y1 )10 ⊕ (x1 y2 − x2 y1 )14 ,

⊕ (x1 y1 + iρx2 y2 )11 ⊕ (x1 y1 − iρx2 y2 )15 , 22

(F.67)



x2 y2 −x1 y1

x2 y1 x1 y2

(F.66)

21

⊕ (x1 y2 + ix2 y1 )12 ⊕ (x1 y2 − ix2 y1 )16 , 23

(F.68)



x2 y2 −x1 y1

⊕ (x1 y2 + iρx2 y1 )13 ⊕ (x1 y2 − iρx2 y1 )17 , 21

(F.69)



x2 y2 −ix1 y1

⊕ (x1 y2 + x2 y1 )12 ⊕ (x1 y2 − x2 y1 )16 , 23

(F.70)



⊕ (x1 y2 + iρx2 y1 )13 ⊕ (x1 y2 − iρx2 y1 )17 , 24

(F.71)

F.6 (Z6 × Z2 )  Z2









x1 x2

x1 x2

x1 x2

x1 x2



 ⊗

22



 ⊗

23



 ⊗

23



 ⊗

24

y1 y2

y1 y2

y1 y2

y1 y2

277



 = 24



 = 23



 = 24



 = 24

x2 y2 x1 y1

x2 y2 x1 y1

⊕ (x1 y2 + ix2 y1 )10 ⊕ (x1 y2 − ix2 y1 )14 , 21

(F.72)



x2 y2 ix1 y1

⊕ (x1 y2 + x2 y1 )10 ⊕ (x1 y2 − x2 y1 )14 , 21

(F.73)



⊕ (x1 y2 + ρx2 y1 )11 ⊕ (x1 y2 − ρx2 y1 )15 , 22

x2 y2 −x1 y1

(F.74)

⊕ (x1 y2 + x2 y1 )12 ⊕ (x1 y2 − x2 y1 )16 . 23

(F.75)

The tensor products between doublets and singlets are  (x)1i ⊗

y1 y2



 =

2j

xy1 ρ i xy2

. 2i+j

(F.76)

mod 4

The tensor products between singlets are 1k ⊗ 1k = 1k+k 

(F.77)

mod 8 .

F.6 (Z6 × Z2 )  Z2

Now we consider the group (Z6 × Z2 )  Z2 . We denote the generator of Z6 by a, while the first and second Z2 generators are written b and c, respectively. The generators a, b, and c satisfy the conditions a 6 = e, c−1 ab = a 5 ,

b2 = e,

c2 = e,

c−1 bc = a 3 b,

ab = ba.

(F.78)

278

F Other Smaller Groups

Table F.7 Characters of (Z6 × Z2 )  Z2 h

χ1++

χ1+−

χ1−+

χ1−−

χ2 1

χ 22

χ 23

χ 24

χ 25

1

1

1

1

1

2

2

2

2

C1

(1)

2

1

1

1

1

2

2

−2

−2

−2

C2(1)

6

1

1

1

1

−1

−1

1

1

−2

C2

(2)

3

1

1

1

1

−1

−1

−1

−1

2

C2(3)

2

1

1

−1

−1

2

−2

0

6

1

1

−1

−1

−1

1

C2(5) C6(1) C6(2)

6

1

1

−1

−1

−1

1

3i √ − 3i

√ − 3i √ 3i

0

C2(4)

0 √

0

2

1

−1

1

−1

0

0

0

0

0

6

1

−1

−1

1

0

0

0

0

0

C1

2

0

All elements can be written in the form a k b cm with k = 0, . . . , 5, and , m = 0, 1. These elements are classified into nine conjugacy classes: C1 : C1(1) (1) C2 (2) C2 C2(3) (4) C2 (5) C2 (1) C6 (2) C6

: : : : : : : :

{e},  3 a ,   a, a 5 ,  2 4 a ,a ,   b, a 3 b ,   ab, a 2 b ,  4  a b, a 5 b ,   c, ac, a 2 c, a 3 c, a 4 c, a 5 c ,   bc, abc, a 2 bc, a 3 bc, a 4 bc, a 5 bc ,

h = 1, h = 2, h = 2, h = 2, h = 2,

(F.79)

h = 6, h = 6, h = 2, h = 6.

The group (Z6 × Z2)  Z2 has four singlets denoted by 1±± and five doublets 2k with k = 1, . . . , 5. The characters are shown in Table F.7. For singlets, the generators b and c can be represented by b = ±1 on 1±s ,

(F.80)

c = ±1 on 1s± ,

(F.81)

for both s = ±, and

for both s = ±, while the generator a is represented by a = 1 on all the singlets. For doublets, we use the following representations for the generators a and b:  a=

ω 0

0 , ω2

 b=

1 0 , 0 1

on 21 ,

(F.82)

F.6 (Z6 × Z2 )  Z2

279

 a= 

0 , ω2

ω 0



−1 0 , on 22 , 0 −1  1 0 b= , on 23 , 0 −1  1 0 b= , on 24 , 0 −1  1 0 b= , on 25 , 0 −1

b=

0 a= , −ω  −ω 0 , a= 0 −ω2  −1 0 a= , 0 −1 −ω2 0

(F.83) (F.84) (F.85) (F.86)

where ω = e2πi/3 , while c is represented by  c=

0 1

1 , 0

(F.87)

on all the doublets. The tensor products between doublets are: 





x1 x2

x1 x2

x1 x2





21

y ⊗ 1 y2



 ⊗ 21



 ⊗ 21

y1 y2

y1 y2

 =

21



 =

22



 =

23

x2 y 2 x1 y1

x2 y 2 x1 y1

x2 y 1 x1 y2

21

⊕ (x1 y2 + x2 y1 )1++ ⊕ (x1 y2 − x2 y1 )1+− , (F.88)

22

⊕ (x1 y2 + x2 y1 )1−+ ⊕ (x1 y2 − x2 y1 )1−− ,



 ⊕ 24

x1 y1 x2 y2

(F.89)

,

(F.90)

25

  y1 x1 y 1 x2 y1 ⊗ = ⊕ , (F.91) y2 2 x2 y2 2 x1 y2 2 21 4 3 5     x1 y1 x2 y 1 x1 y1 ⊗ = ⊕ , (F.92) x2 2 y2 2 x1 y2 2 x2 y2 2 1 5 3 4    x1 y x2 y 2 ⊗ 1 = ⊕ (x1 y2 + x2 y1 )1++ ⊕ (x1 y2 − x2 y1 )1+− , x2 2 y2 2 x1 y1 2 

x1 x2







2

 

x1 x2 x1 x2

2



 ⊗



22

 ⊗

22

y1 y2 y1 y2

1



 =

23



 =

24

x1 y 2 x2 y1 x2 y 2 x1 y1



 ⊕



23

 ⊕

24

x2 y2 x1 y1 x1 y2 x2 y1

(F.93)

,

(F.94)

,

(F.95)

25



25

280

F Other Smaller Groups

 



 

 

x1 x2 x1 x2

x1 x2

x1 x2 x1 x2

x1 x2 x1 x2



 ⊗



22

 ⊗

23



⊗ 23



 ⊗



23

 ⊗

24



⊗

24

25

y1 y2 y1 y2

y1 y2

y1 y2 y1 y2

y1 y2



y ⊗ 1 y2



 =

25





= 23



= 24



 =

25





= 24



= 25

x2 y 2 x1 y1 x1 y 1 x2 y2

x1 y 2 x2 y1

x2 y 2 x1 y1 x2 y 1 x1 y1

x1 y 1 x2 y2



23

21

x1 y2 x2 y1

(F.96)

, 24

⊕ (x1 y2 + x2 y1 )1−+ ⊕ (x1 y2 − x2 y1 )1−− , (F.97)

22

⊕ (x1 y1 + x2 y2 )1++ ⊕ (x1 y1 − x2 y2 )1+− ,



 ⊕



21

21

x2 y1 x1 y2

(F.98)



(F.99)

, 22

⊕ (x1 y2 + x2 y1 )1−+ ⊕ (x1 y2 − x2 y1 )1−− , 





25

 ⊕

⊕ 21

x1 y2 x2 y1

(F.100)



(F.101)

, 22

= (x1 y1 + x2 y2 )1++ ⊕ (x1 y1 − x2 y2 )1+− ⊕ (x1 y2 + x2 y1 )1−+ ⊕ (x1 y2 − x2 y1 )1−− .

(F.102)

The tensor products between doublets and singlets are:   y xy1 (x)1+± ⊗ 1 = , (F.103) y2 2 ±xy2 2 i i     y xy1 y xy1 = , (x)1−± ⊗ 1 = , (x)1−± ⊗ 1 y2 2 ±xy2 2 y2 2 ±xy2 2 1

 (x)1−± ⊗  (x)1−± ⊗

y1 y2

y1 y2

2



 =

23



= 25

xy2 ±xy1

xy2 ±xy1

2



 , 24

(x)1−± ⊗

y1 y2

1



 = 24

xy2 ±xy1



(F.104) , 23

(F.105)

.

(F.106)

25

The tensor products between singlets are 1s1 s2 ⊗ 1s1 s2 = 1s1 s2 , where s1 = s1 s1 and s2 = s2 s2 .

(F.107)

F.7 Z9  Z3

281

F.7 Z9  Z3 We denote the Z9 and Z3 generators by a and b, respectively. They satisfy a 9 = 1,

ab = ba 7 .

(F.108)

Using these, all elements of Z9  Z3 can be written in the form g = bm a n ,

(F.109)

with m = 0, 1, 2, and n = 0, . . . , 8. These elements are classified into eleven conjugacy classes: C1 : C1(2) (3) C1 (1) C3 C3(2) (3) C3 (4) C3 (5) C3 C3(6) (7) C3 (8) C3

: : : : : : : : : :

{e}, h = 1,  3 h = 3, a ,  6 h = 3, a ,   3 6 h = 3, b, ba , ba ,   4 7 h = 9, ba, ba , ba ,   2 5 8 h = 9, ba , ba , ba ,  2 2 3 2 6 h = 3, b ,b a ,b a ,   2 b a, b2 a 4 , b2 a 7 , h = 9,  2 2 2 5 2 8 b a , b a , b a , h = 9,   h = 3, a, a 4 , a 7 ,  2 5 8 h = 3. a ,a ,a ,

(F.110)

This group has nine singlets 1n,k with n, k = 0, 1, 2, and two triplets 31 and 32 . The characters are shown in Table F.8, where ω = e2πi/3 . The generators a and b are represented by a = ωk ,

b = ω ,

on 1k, .

We use the following representations for the generator a: ⎛ ⎛ 2 ⎞ ⎞ ρ 0 0 ρ 0 0 a = ⎝ 0 ρ 4 0 ⎠ , on 31 , a = ⎝ 0 ρ8 0 ⎠ , 7 0 0 ρ 0 0 ρ5 where ρ = e2πi/9 , while b is represented by ⎛ ⎞ 0 1 0 b = ⎝0 0 1⎠, 1 0 0

(F.111)

on 32 ,

(F.112)

(F.113)

282

F Other Smaller Groups

Table F.8 Characters of T9 h

χ100

χ101

χ102

χ110

χ111

χ112

χ120

χ121

χ122

χ 31

C1

1

1

1

1

1

1

1

1

1

1

3

3

C1(2)

3

1

1

1

1

1

1

1

1

1

3ω

3ω2

C1(2)

3

1

1

1

1

1

1

1

1

1

3ω2

3ω

1

ω

ω2

1

ω

ω2

0

0

(1) C3 C3(2) (3) C3 C3(4) (5) C3 C3(6) (7) C3 C3(8)

χ 32

9

1

ω

ω2

9

1

ω

ω2

ω

ω2

1

ω2

1

ω

0

0

3

1

ω

ω2

ω2

1

ω2

ω

ω2

1

0

0

9

1

ω2

ω

1

ω2

ω

1

ω2

ω

0

0

9

1

ω2

ω

ω

1

ω2

ω2

ω

1

0

0

3

1

ω2

ω

ω2

ω

1

ω

1

ω2

0

0

ω2

ω2

0

0

ω

ω

0

0

3

1

1

1

ω

ω

ω

ω2

3

1

1

1

ω2

ω2

ω2

ω

on both triplets. The tensor products between triplets are: ⎞ ⎞ ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ ⎛ x1 y1 x1 y 1 x2 y3 x3 y2 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x2 y2 ⎠ ⊕ ⎝ x3 y1 ⎠ ⊕ ⎝ x1 y3 ⎠ , (F.114) x3 3 y3 3 x3 y3 3 x1 y2 3 x2 y1 3 1 1 2 2 2 ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎛ ⎞ y1 x3 y 3 x1 y2 x2 y1 x1 ⎝ x2 ⎠ ⊗ ⎝ y2 ⎠ = ⎝ x1 y1 ⎠ ⊕ ⎝ x2 y3 ⎠ ⊕ ⎝ x3 y2 ⎠ , (F.115) x3 3 y3 3 x2 y2 3 x3 y1 3 x1 y3 3 2 2 1 1 1 ⎛ ⎞ ⎛ ⎞ x1 y1    ⎝ x2 ⎠ ⊗ ⎝ y 2 ⎠ = x1 y2 + ω2k x2 y3 + ωk x3 y1 1 0k x3 3 y3 3 k=0,1,2 1 2    x1 y1 + ω2k x2 y2 + ωk x3 y3 1 ⊕ ⎛

1k

k=0,1,2

⊕

   x1 y3 + ω2k x2 y1 + ωk x3 y2 1 . (F.116) 2k

k=0,1,2

The tensor products between triplets and singlets are: ⎛

(x)10k

⎛ ⎞ ⎞ xy1 y1 ⊗ ⎝ y2 ⎠ = ⎝ ωk xy2 ⎠ , y3 3 ω2k xy3 3 1

1

⎛

(x)10k

⎛ ⎞ ⎞ xy1 y1 ⊗ ⎝ y2 ⎠ = ⎝ ωk xy2 ⎠ , y3 3 ω2k xy3 3 2

2

(F.117)

References

283

⎛

(x)11k

⎛ ⎞ ⎞ xy3 y1 ⊗ ⎝ y2 ⎠ = ⎝ ωk xy1 ⎠ , y3 3 ω2k xy2 3 1

⎛

(x)11k

2

1

⎛

(x)12k

⎛ ⎞ ⎞ xy2 y1 ⊗ ⎝ y2 ⎠ = ⎝ ωk xy3 ⎠ , y3 3 ω2k xy1 3 1

1

⎛ ⎞ ⎞ xy2 y1 ⊗ ⎝ y2 ⎠ = ⎝ ωk xy3 ⎠ , y3 3 ω2k xy1 3 ⎛

(x)12k

2

(F.118) ⎞

⎛ ⎞ xy3 y1 ⊗ ⎝ y2 ⎠ = ⎝ ωk xy1 ⎠ . y3 3 ω2k xy2 3 2

2

(F.119) The tensor products between singlets are 1n,k ⊗ 1n ,k  = 1n+n ,k+k  .

(F.120)

References 1. Frampton, P.H., Kephart, T.W.: Int. J. Mod. Phys. A 10, 4689 (1995). arXiv:hep-ph/9409330 2. Frampton, P.H., Kephart, T.W., Rohm, R.M.: Phys. Lett. B 679, 478 (2009). arXiv:0904.0420 [hep-ph]

Index

A Abelian, 14 A4 , 31 A4 flavor model, 207 A5 , 34 Alternating group, 31 Anomaly of discrete symmetry, 187 Associativity, 13 B Bimaximal mixing, 219, 221 Binary dihedral group, 61 Breaking pattern, 147 C Character, 16 Closure, 13 Completely reducible, 16 Conjugacy class, 16 Cube, 25 Cummins-Patera’s basis, 35 Cyclic group, 13 D D4 , 58 D5 , 59 Δ(27), 94 Δ(54), 138 Dihedral group, 51 Dimension, 16 Dodecahedron, 34 Double covering group, 43 F Flavor mixing, 205 Fujikawa’s method, 185

G Golden ratio, 219, 220 Gravitational anomaly, 186 H Homomorphic, 16 I Icosahedron, 34 Invariant subspace, 16 Irreducible, 16 Isomorphic, 16 L Lagrange’s theorem, 14, 229 N Non-Abelian, 14 Normal subgroup, 14 O Octahedron, 25 Order, 14 Origin of flavor symmetries, 223 P Pentagon, 59 Polygon, 51 Q Q4 , 66 Q6 , 67 QD16 , 72 R Reducible, 16 Representation, 16

H. Ishimori et al., An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists, Lecture Notes in Physics 858, DOI 10.1007/978-3-642-30805-5, © Springer-Verlag Berlin Heidelberg 2012

285

286 S Schur’s lemma, 230 S3 , 21 S4 , 25 S4 flavor model, 211 Semi-direct product, 19 Shirai’s basis, 35, 37 Square, 58 Subgroup, 15, 147 Symmetric group, 14, 21 Σ(18), 78

Index Σ(32), 80 Σ(50), 84 Σ(81), 113 T Tetrahedron, 31, 32 T  , 43 T7 , 100 T13 , 102 T19 , 104 Tri-bimaximal mixing, 205, 206