Polynomial Identities And Combinatorial Methods

Polynomial Identities and Combinatorial Methods

edited by

Antonio Giambruno University of Palermo Palermo, Italy

Amitai Regev The Weizmann Institute of Science Rehovot, Israel

Mikhail Zaicev Moscow State University Moscow, Russia

>>

MARCEL DEKKER, INC.

D E K K E R

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

NEW YORK • BASEL

Although great care has been taken to provide accurate and current information, neither the author(s) nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. The material contained herein is not intended to provide specific advice or recommendations for any specific situation. Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-4051-3 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016, U.S.A. tel: 212-696-9000; fax: 212-685-4540 Distribution and Customer Service Marcel Dekker, Inc. Cimarron Road, Monticello, New York 12701, U.S.A. tel: 800-228-1160; fax: 845-796-1772 Eastern Hemisphere Distribution Marcel Dekker AC Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universitat Siegen

W. S. Massey Yale University

TM

Anil Nerode Cornell University

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

N. Jacobson, Exceptional Lie Algebras L.-A. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis /. Satake, Classification Theory of Semi-Simple Algebraic Groups F. Hirzebruch et a/., Different/able Manifolds and Quadratic Forms I. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Burckel, Characterization of C(X) Among Its Subalgebras B. R. McDonald et a/., Ring Theory Y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus et a/., Calkin Algebras and Algebras of Operators on Banach Spaces E. O. Roxin et a/., Differential Games and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thornier, Topology and Its Applications J. M. Lopez and K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carmeli and S. Matin, Representations of the Rotation and Lorentz Groups G. B. Seligman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis L. Cesari et a/., Nonlinear Functional Analysis and Differential Equations J. J. Schaffer, Geometry of Spheres in Normed Spaces K. YanoandM. Kon, Anti-Invariant Submanifolds W. V. Vasconcelos, The Rings of Dimension Two R E. Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology S. K. Jain, Ring Theory B. R. McDonald and R. A. Morris, Ring Theory II R. B. Mura and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynamical Systems H.-C. Wang, Homogeneous Branch Algebras E. O. Roxin et a/., Differential Games and Control Theory II R D. Porter, Introduction to Fibre Bundles M. Altman, Contractors and Contractor Directions Theory and Applications J. S. Golan, Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis R Gordon, Representation Theory of Algebras M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. I. Arruda eta/., Mathematical Logic F. Van Oystaeyen, Ring Theory F. Van Oystaeyen and A. Verschoren, Reflectors and Localization M. Satyanarayana, Positively Ordered Semigroups D. L Russell, Mathematics of Finite-Dimensional Control Systems P.-T. Liu and E. Roxin, Differential Games and Control Theory III A. Geramita and J. Seberry, Orthogonal Designs J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics C. Byrnes, Partial Differential Equations and Geometry G. Klambauer, Problems and Propositions in Analysis J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields F. Van Oystaeyen, Ring Theory B. Kadem, Binary Time Series J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems R. L Stemberg et a/., Nonlinear Partial Differential Equations in Engineering and Applied Science B. R. McDonald, Ring Theory and Algebra III J. S. Golan, Structure Sheaves Over a Noncommutative Ring T. V. Narayana et a/.. Combinatorics, Representation Theory and Statistical Methods in Groups T. A. Burton, Modeling and Differential Equations in Biology K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory

47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120.

TM

J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces O. A. Nielson, Direct Integral Theory J. E. Smith et a/., Ordered Groups J. Cronin, Mathematics of Cell Electrophysiology J. W. Brewer, Power Series Over Commutative Rings P. K. Kamthan and M. Gupta, Sequence Spaces and Series T. G. McLaughlin, Regressive Sets and the Theory of Isols T. L Herdman et a/., Integral and Functional Differential Equations R. Draper, Commutative Algebra W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules sentations of Simple Lie Algebras R. L. Devaney and 2. H. Nitecki, Classical Mechanics and Dynamical Systems J. Van Gee/, Places and Valuations in Noncommutative Ring Theory C. Faith, Injective Modules and Injective Quotient Rings A. Fiacco, Mathematical Programming with Data Perturbations I P. Schultz et a/., Algebraic Structures and Applications L Bican et a/., Rings, Modules, and Preradicals D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces C.-C. Yang, Factorization Theory of Meromorphic Functions O. Taussky, Ternary Quadratic Forms and Norms S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications K. B. Hannsgen et a/., Volterra and Functional Differential Equations N. L. Johnson et a/., Finite Geometries G. /. Zapata, Functional Analysis, Holomorphy, and Approximation Theory S. Greco and G. Valla, Commutative Algebra A. V. Fiacco, Mathematical Programming with Data Perturbations II J.-B. Hiriart-Urrutyetal., Optimization A. Figa Talamanca and M. A. Picarde/lo, Harmonic Analysis on Free Groups M. Harada, Factor Categories with Applications to Direct Decomposition of Modules V. I. Istratescu, Strict Convexity and Complex Strict Convexity V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations H. L. Manocha and J. B. Srivastava, Algebra and Its Applications D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Problems J. W. Longley, Least Squares Computations Using Orthogonalization Methods L. P. de Alcantara, Mathematical Logic and Formal Systems C. E. Au/l, Rings of Continuous Functions R. Chuaqui, Analysis, Geometry, and Probability L. Fuchs and L. Salce, Modules Overvaluation Domains P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics W. B. Powell and C. Tsinakis, Ordered Algebraic Structures G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, Applications R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications J. H. Lightbourne III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Equations C. A. Baker and L. M. Batten, Finite Geometries J. W. Brewer et a/., Linear Systems Over Commutative Rings C. McCrory and T. Shifrin, Geometry and Topology D. W. Kueke et a/.. Mathematical Logic and Theoretical Computer Science B.-L Lin and S. Simons, Nonlinear and Convex Analysis S. J. Lee, Operator Methods for Optimal Control Problems V. Lakshmikantham, Nonlinear Analysis and Applications S. F. McCormick, Multigrid Methods M. C. Tangora, Computers in Algebra D. V. Chudnovsky and G. V. Chudnovsky, Search Theory D, V. Chudnovsky and R. D. Jenks, Computer Algebra M. C. Tangora, Computers in Geometry and Topology P. Nelson et a/., Transport Theory, Invariant Imbedding, and Integral Equations P. Clement et a/., Semigroup Theory and Applications J. Vinuesa, Orthogonal Polynomials and Their Applications C. M. Dafermos et a/., Differential Equations £. O. Roxin, Modern Optimal Control J. C. Diaz, Mathematics for Large Scale Computing

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

for Repre-

Arithmetic

and Their

Differential

121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183.

TM

P. S. Milojevft Nonlinear Functional Analysis C. Sadosky, Analysis and Partial Differential Equations R. M. Shortt, General Topology and Applications R. Wong, Asymptotic and Computational Analysis D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics W. D. Wallis et al., Combinatorial Designs and Applications S. Elaydi, Differential Equations G. Chen etal.. Distributed Parameter Control Systems W. N. Everitt, Inequalities H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differential Equations O. Anno etal., Mathematical Population Dynamics S. Coen, Geometry and Complex Variables J. A. Goldstein et al., Differential Equations with Applications in Biology, Physics, and Engineering S. J. Andima et al., General Topology and Applications P Clement et al., Semigroup Theory and Evolution Equations K. Jarosz, Function Spaces J. M. Bayod et al., p-adic Functional Analysis G. A. Anastassiou, Approximation Theory R. S. Rees, Graphs, Matrices, and Designs G. Abrams et al., Methods in Module Theory G. L. Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing M. C. JoshiandA. V. Balakrishnan, Mathematical Theory of Control G. Komatsu and Y. Sakane, Complex Geometry /. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations T. Mabuchiand S. Mukai, Einstein Metrics and Yang-Mills Connections L Fuchs and R. Gobel, Abelian Groups A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum G. Dore et a/., Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedt et a/., Functional Analysis K. G. Fischer etal., Computational Algebra K. D. Elworthyetal., Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, et a/., Commutative Ring Theory S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions P. Clement and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research W. O. Bray et a/., Fourier Analysis J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pavel, Optimal Control of Differential Equations M. Ikawa, Spectral and Scattering Theory X. Liu and D. Siegel, Comparison Methods and Stability Theory J.-P. Zolesio, Boundary Control and Variation M. Kfizeketal., Finite Element Methods G. Da Prato and L Tubaro, Control of Partial Differential Equations E. Ballico, Projective Geometry with Applications M. Costabeletal., Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twistor Theory H. Cooket a/., Continua D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et al., Complex Analysis and Geometry E. Casas, Control of Partial Differential Equations and Applications N. Kalton et al.. Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et al., Differential Equations and Control Theory P. Marcelliniet al. Partial Differential Equations and Applications A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type M. Maruyama, Moduli of Vector Bundles A. Ursini and P. Agliano, Logic and Algebra X. H. Cao et al., Rings, Groups, and Algebras D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235.

J. E. Andersen et at., Geometry and Physics P.-J. Cahen et al., Commutative Ring Theory J. A. Goldstein et al.. Stochastic Processes and Functional Analysis A. Sorbi, Complexity, Logic, and Recursion Theory G. Da Prato and J.-P. Zolesio, Partial Differential Equation Methods in Control and Shape Analysis D. D. Anderson, Factorization in Integral Domains N. L. Johnson, Mostly Finite Geometries D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm-Liouville Problems W. H. Schikhofet a/., p-adic Functional Analysis S. Sertdz, Algebraic Geometry G. Can'sti and E. Mitidieri, Reaction Diffusion Systems A. V. Fiacco, Mathematical Programming with Data Perturbations M. Krizek et al., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups V. Drensky et a/., Methods in Ring Theory W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions P. E. Newstead, Algebraic Geometry D. Dikranjan and L Salce, Abelian Groups, Module Theory, and Topology Z. Chen et a/., Advances in Computational Mathematics X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs C. Y. Yildirim and S. A. Stepanov, Number Theory and Its Applications D. E. Dobbs et a/., Advances in Commutative Ring Theory F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry J. Kakol et a/., p-adic Functional Analysis M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of Algebras R. Costa et al., Nonassociative Algebra and Its Applications T.-X. He, Wavelet Analysis and Multiresolution Methods H. Hudzik and L Skrzypczak, Function Spaces: The Fifth Conference J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis G. Lumerand L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnol et al.. Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. AH Mehmeti et al.. Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra A. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis R. Salvi, The Navier-Stokes Equations F. U. Coe/ho and H. A. Merk/en, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Camiellietal., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vail/ant, Hyperbolic Differential Operators A. Giambruno et al., Polynomial Identities and Combinatorial Methods G. R. Goldstein etal., Evolution Equations

Additional Volumes in Preparation

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

Preface

This volume contains the proceedings of the conference on Polynomial Identities and Combinatorial Methods, held on the island of Pantelleria, Italy. It was the fourth in a series of meetings in the last decade concerning the theory of associative and nonassociative algebras satisfying polynomial identities (Pi-algebras). The first of these meetings was a small workshop in Palermo, Italy, in 1992. The second was the conference entitled Methods in Ring Theory, held in 1997 in Trento, Italy. The proceedings of that conference were published in the Marcel Dekker series Lecture Notes in Pure and Applied Mathematics, Volume 198, and it is now a standard reference for specialists working in the area of polynomial identities. Considerable progress could be observed in the theory of algebras with polynomial identities during the years following the Trento conference. Some of the most important achievements in this area were due to a combination of algebraic techniques with analytical and combinatorial methods in the study of various numerical characteristics of Pi-algebras and their identities. Leading specialists in this area met in Rehovot, Israel, in May 2000, in the Workshop on Growth Phenomena in Associative and Lie Pi-algebras. Unfortunately, the results presented in Rehovot were not collected in one volume. To make up for this deficiency we hereby offer the proceedings of the Pantelleria conference, presenting the up-to-date status and tendencies in this area. The conference featured the latest results in the theory of polynomial identities and a presentation of different methods and techniques pertaining to different areas, such as algebraic combinatorics, invariant theory, and representation theory, both of the symmetric and the classical groups, and of Lie algebras and superalgebras. During the conference one-hour invited lectures were given by Y. Bahturin, A. Belov, O. Di Vincenzo, M. Domokos, V. Drensky, E. Formanek, A. Giambruno, A. Guterman, P. Koshlukov, S. Mishchenko, V. Petrogradsky, C. Procesi, A. Regev, L. H. Rowen, I. Shestakov, and M. Zaicev. In addition, several other invited talks of shorter lengths were presented. This volume includes the papers of most of the principal speakers and some other invited contributions related to the conference. Even though the contents of this volume cover a broad range of themes, from ring theory to combinatorics to invariant theory, they still have a common thread in the theory of polynomial identities. The book will be useful to all researchers working with polynomial identities, varieties of associative algebras, Lie and Liebnitz algebras, and their generalizations. It will also be of interest to specialists in free algebras, growth functions of algebras, and, more generally, to mathematicians who apply numerical and analytical methods in algebra.

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

The editors wish to express their appreciation to the following agencies and institutions that have contributed financial support: the Gruppo Nazionale per le Strutture Algebriche Geometriche ed Applicazioni of the Istituto Nazionale di Alta Matematica, the national research project Algebre con Identita Polinomiali of the MURST, and the University of Palermo. Antonio Giambruno Amitai Regev Mikhail Zaicev

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

Contents

Preface Contributors

TM

1.

Linearization Method of Computing Z2-Codimensions of Identities of the Grassmann Algebra N. Anisimov

2.

Cocommutative Hopf Algebras Acting on Quantum Polynomials and Their Invariants Vyacheslav A. Artamonov

3.

Combinatorial Properties of Free Algebras of Schreier Varieties Vyacheslav A. Artamonov, Alexander A. Mikhalev, and Alexander V. Mikhalev

4.

Graded Algebras and Graded Identities Yu. A. Bahturin and M. V. Zaicev

5.

Computational Approach to Polynomial Identities of Matrices: A Survey Francesco Benanti, James Demmel, Vesselin Drensky, and Plamen Koev

6.

Poincare Series of Generic Matrices Allan Berele

1.

Combinatorial Methods for the Computation of Trace Cocharacters Luisa Carini

8.

Polynomial Identities for Graded Algebras Onofrio Mario Di Vincenzo

9.

Matrix Invariants and the Failure of Weyl's Theorem M. Domokos

10.

Free Nilpotent-by-Abelian Leibniz Algebras Vesselin Drensky and Giulia Maria Piacentini Cattaneo

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

TM

11.

Pebbles and Expansions in the Polynomial Ring Adriano M. Garsia

12.

Group Actions, Codimensions, and Exponential Behaviour A. Giambnmo

13.

Monotone Matrix Maps Preserve Non-maximal Rank A. Guterman

14.

Explicit Decompositions of the Group Algebras FSn and FAn A. Henke and A. Regev

15.

Graded and Ordinary Polynomial Identities in Matrix and Related Algebras Plamen Koshlukov

16.

Varieties of Linear Algebras with Almost Polynomial Growth S. P. Mishchenko

17.

Algebras with Involution, Superalgebras, and Proper Subvarieties Vincenzo Nardozza

18.

Gradings and Graded Identities of the Algebra of n x n Upper Triangular Matrices Angela Valenti

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

Contributors

N. Anisimov, Chair of Higher Algebra, Faculty of Mechanics and Mathematics, Moscow State University, 119899, Moscow, Russia E-mail: [email protected] Vyacheslav A. Artamonov, Department of Algebra, Faculty of Mechanics and Mathematics, Moscow State University, Leninsky Gory, 119992, GSP-2, Moscow, Russia E-mail: [email protected] Yu. A. Bahturin, Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NF, A1A 5K9, Canada, and Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992, Russia E-mail: [email protected] ca Francesca Benanti, Dipartimento di Matematica ed Applicazioni, Universita di Palermo, via Archirafi 34, 90123 Palermo, Italy E-mail: [email protected] Allan Berele, Department of Mathematics, DePaul University, Chicago, IL 60614 USA E-mail: [email protected] Luisa Carini, Dipartimento di Matematica ed Applicazioni, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy E-mail: [email protected] James Demmel, Department of Mathematics, Computer Science Division, University of California at Berkeley, Berkeley, CA 94720, USA E-mail: [email protected] Onofrio Mario Di Vincenzo, Dipartimento Interuniversitario di Matematica, Universita degli Studi di Bari, via Orabona 4, 70125 Bari, Italy E-mail: [email protected]

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

Matyas Domokos. Renyi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary E-mail: [email protected] Temporary address (until February 2004): Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland E-mail: [email protected] Vesselin Drensky, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Akad. G. Bonchev Str., Block 8,1113 Sofia, Bulgaria E-mail: [email protected] Adriano M. Garsia, Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112 USA E-mail: [email protected] Antonio Giambruno, Dipartimento di Matematica ed Applicazioni, Palermo, Via Archirafi 34, 90123 Palermo, Italy E-mail: [email protected]

Universita di

A. Guterman, Department of Higher Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119899, Russia E-mail: [email protected] A. Henke, Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, England E-mail: [email protected] Plamen Koev, Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA E-mail: [email protected] Plamen Koshlukov, IMECC, UNICAMP, Cx.P. 6065, 13083-970 Campinas, SP, Brazil E-mail: [email protected] Alexander A. Mikhalev, Department of Mechanics and Mathematics, Moscow State University, 119899, Moscow, Russia E-mail: [email protected] Sergey P. Mishchenko, Department of Algebra and Geometric Computations, Faculty of Mathematics and Mechanics, Ul'yanovsk State University, Ul'yanovsk, 432700, Russia E-mail: [email protected] Vincenzo Nardozza, Dipartimento di Matematica, Universita della Basilicata, C.da Macchia Romana, 85100 Potenza, Italy E-mail: [email protected]

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

Ciulia Maria Piacentini Cattaneo, Department of Mathematics, University of Rome "Tor Vergata", Via della Ricerca Scientifica, 00133 Rome, Italy E-mail: [email protected] Amitai Regev, Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel E-mail: [email protected] Angela Valenti, Dipartimento di Matematica ed Applicazioni, Universita di Palermo, Via Archirafi 34, 90123 Palermo, Italy E-mail: [email protected] M. V. Zaicev, Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, 119992, Russia E-mail: [email protected]

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

Linearization method of computing Z2 codimensions of identities of the Grassmann algebra N. ANISIMOV Chair of Higher Algebra, Faculty of Mechanics and Mathematics, Moscow State University, 119899, Moscow, Russia. E-mail: [email protected]

1

INTRODUCTION

Let A be an associative algebra over a field F of characteristic 0, and let G be a finite group of automorphisms and anti-automorphisms of A. Also let X = {.TI, x,2, • • •} be a countable set of indeterminates. The free algebra with the set of free generators < X\G >= {xf\i £ N,# G G} is called the algebra of G-polynomials. The space

is called the space of multilinear G-polynomials in the variables xi,...,xn. The polynomial f(xi,...,xn,G) €E Vn(x,G) is called an n-linear Gidentity if for arbitrary elements 0,1,.... an of A, /(GI, . . . , an, G) = 0; here we use the notation a9 — g(a) for all a G A, g € G. All n-linear G-identities form the ideal Idn(A, G) of n-linear G-identities. The sequence Cn(A, G) = dirnp—— is called the sequence of G-codimensions of the ideal of the G-identities of the algebra A. If the group G is generated by an element 1.1 < i\ < .. . < i^} form a basis D\ of A. There is a natural Z2-grading on A, A = AO © AI, where AO and AI are spanned by the basic monomials of even and of odd length correspondingly. The Z2~ graded elements of A commute by the following rule: if g € A,; and h £ Aj, then gh = ( — l)ljhg. The length a of the monomial a 6 D\ is the number of generators in a: je,;, ... Ci k = A;. Let (p be an automorphism of order two on the algebra A. If it's action on the generators is defined bv the formula

then the mapping defined on the generators by the formula

3

and homomorphically continued on A is also an automorphism of order two on the Grassmann algebra. The automorphism tpi is a linear operator in

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

the space L — span^{ei, £2, . . .}. In further considerations we will assume without loss of generality that the generators of the Grassmann algebra form the eigenbasis for the operator (pi in the space L. The subspaces corresponding to the eigenvalues 1 and — 1 are denoted by L\ and L-\.

2

Zs-IDENTITIES OF THE GRASSMANN ALGEBRA

First we note that for an arbitrary automorphism (p of the Grassmann algebra A with dim LI = dimL_i = oo the c^-codimensions of the identities of A were computed in [4]. But cn(A, k, at least one of the chosen monomials a^,..., a,ik is of a non-unit length. Since (p is a graded automorphism we may assume, without loss of generality, that a^ = e^aj- is of non-unit length. Then

TM

Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

Since aJ:J + X1 1=2 av I = "?• — 1 < "?• therefore by induction

Since 9i — •"•• V^(j;, ^) by the equality r(a;^) = xfl and prove that if (r/) € Id n (A, ^) then / 6 Id n (A,an)

\

(n

= 9j

and may be computed by the formula

In [4] it was proved that (r/) € Id n (A, (^) if and only if

E

(//("V)