-F'ublications Mathematical Research Institute 6
Representat Theoryof Flnlte Groups Editedby R.Solomon
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Topology '90, B. Apanasov, W D. Neumann, A. W Reid, L. Siebenmann (Eds.) The Arithmetic of Function Fields, D. Goss, D. R. Ha.ves, M. 1. Rosen (Eds.) Geometric Group Theory, R. Charney, M. Davis, M. Shapiro (Eds.) Groups, Difference Sets, and the Monster, K. T Arasu, 1. F Dillon, K. Harada, S. Sehgal, R. Solomon (Eds.) Convergence in Ergodic Theory and Probability, V Bergelson, P March, 1. Rosenblatt (Eds.)
Representation Theory of Finite Groups Proceedings of a Special Research Quarter at The Ohio State University, Spring 1995
Editor Ronald Solomon
Waiter de Gruyter . Berlin· New York 1997
Editor RONALD SOLOMON
Department of Mathematics, The Ohio State University 231 West 18th Avenue, Co1umbus, OH 43210, USA Series Editors: Gregory R. Baker Department of Mathematics. The Ohio State University, Columbus, Ohio 43210-1174. USA Karl Rubin Department of Mathematics. Stanford University. Stanford. CA 94305-2125. USA Waiter D. Neumann Department of Mathematics. The Uniwrsity of Melbourne. Parkville. VIC 3052. Australia 1991 IvIathematics Suhject Classification: Primary: 20C Secondary: 20C15. 20C20, 20C30. 20C33 Kcnl"(lrds: Finite group, module, representation. modular representation, character. group algebra. module category
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Representation theory of finite groups: proceedings of a special research quarter at the Ohio State University. spring. 1995 / editor. Ronald Solomon. p. cm. - (Ohio State University Mathematical Research Institute publications. ISSN 0942-0363 : 6) Includes bibliographical references. ISBN 3-II-OI5806-X (alk. paper) I. Finite groups - Congresses. 2. Representations of groups - Congresses. r. Solomon. R. C. (Ronald C.) n. Series. QAI77.R46 1997 5I2'.2-dc21 97-35939 CIP
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Representation theory of finite groups: proceedings of a special research quarter at The Ohio State University, spring 1995/ ed. Ronald Solomon. - Berlin: New York: de Gruyter. 1997 (Ohio State University, Mathematical Research Institute publications: 6) . ISBN 3-II-OI5806-X
© Copyright 1997 by Waiter de Gruyter & Co .. D-I0785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means. electronic or mechanical. including photocopy. recording. or any information storage and retrieval system. without permission in writing from the publisher. Printed in Germany. Typeset using the authors' T EX files: 1. Zimmermann. Freiburg. Printing: Werner Hildebrand. Berlin. Binding: Liideritz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie. Hamburg.
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Preface
In March 1896, Richard Dedekind wrote a letter to Georg Frobenius in which he defined his concept of the "group determinant" of a finite group and stated his results concerning the factorization of this polynomial when the group G is abelian. In a second letter in April, he formulated some conjectures concerning the factorization of the group determinant when G is non-abelian. Frobenius attacked the problem immediately and made substantial progress by Summer 1896. This year 1997 represents the centenary of fundamental papers by three mathematicians - Theodor Molien, Georg Frobenius and WilIiam Burnside each developing many of the foundational results of the theory of complex representations of finite groups. It is fitting then that we now publish this set of papers which gives some measure of the distance this theory has advanced in its first century and some clues as to the roads it will follow in its second century. These proceedings record some of the activity which took place during a special research quarter held at the Ohio State University in Spring 1995. This quarter and the concluding conference were supported by the O.S.U. Mathematical Research Institute and by the National Science Foundation. The single most monumental and important achievement of the theory of group characters is the Odd Order Theorem of Waiter Feit and John G. Thompson. This theorem which formed the entire content of Volume 13, no. 3, of the Pacific Journal of Mathematics, Fall 1963, is a remarkable fusion of the "local structure theory" of finite groups, initiated by Ludvig Sylow and the character theory of finite groups, initiated by Frobenius. And it was our great pleasure at this conference to honor Professor WaIter Feit of Yale University on the occasion of his sixty-fifth birthday. It was a pleasure to bring together many of Waiter's mathematical children and grandchildren, friends and admirers, to celebrate his illustrious career in mathematics. In addition to those who have contributed papers to this volume, many other mathematicians joined in the formal and informal discussions at the conference. We thank all of them for their enthusiastic participation. The speakers, their current affiliations and their topics are:
Preface
VlIl
10nathan L. Alperin, University of Chicago, On endo-permutation modules; Matthew K. Bardoe, Imperial College, University of London, Representations, embeddings and geometries; David G. Benson, University of Georgia, Cohomology of modules for a finite group; Ro.bert Boltje, University of Augsburg, Canonical induction formulae and the defect of a character; Michel BroU(~, Denis-Diderot Universite of Paris, The abelian defect group conjecture infinite reductive groups; Everett Dade, University of Illinois, Urbana-Champaign, Counting characters in blocks; Harald Ellers, Northern Illinois University, On Alperin 's weight conjecture and Brauer's First Main Theorem; Karin Erdmann, University of Oxford, Representations ofsymmetric groups and GLn(K); WaIter Feit, Yale University, Schur indices; Guoqiang Huang, Northern Illinois University, On extended block induction and Brauer's Third Main Theorem; Radha Kessar, Yale University, On blocks and source algebrasfor 2Sn and
2A n
;
Lluis Puig, CNRS, Institut de Mathematiques de Jussieu, On the Morita and Rickard equivalences between Brauer blocks; Geoffrey R. Robinson, University of Leicester, Some open conjectures in block theory; Leonard L. Scott, University of Virginia, On the Lusztig conjectures; Sergei Syskin, Reinsurance Group of America, Locally finite varieties and representations offinite groups; 1. P. Zhang, Peking University, Vertices of irreducible modules.
In conclusion it is a pleasure to acknowledge the efforts of Dr. Radha Kessar, who provided much assistance with the organization and correspon-
Preface
IX
dence for the conference. Also my deep appreciation goes to my wife, Myriam Solomon, who was the guiding hand for the conference reception and dinners and who created the lovely design for the conference coffee mugs, as well as of course tolerating me during the entire period. Many thanks also to Lluis and Isabel Puig, who often acted more as hosts than as guests and who in particular organized a lovely evening of music during the conference, graced by the musical talents of Marcus Linckelmann, Richard Lyons and Isabel Puig. We also grateful acknowledge the re-typing work of Mr. Jwalant Vakil and Ms. Terry England, who put this volume into final polished form. Again thanks to all of the participants and to the O.S.U. Mathematical Research Institute and the National Science Foundation for their generous financial support. Particular thanks to Ann Boyle and Andy Earnest for their efficient handling of our proposal and their kind words of encouragement.
Ron Solomon
Table of Contents
Preface
vii
M. K. Bardoe Embeddings, Geometries and Representations: Connections and Computations
1
D. J. Benson Infinite Dimensional Modules for a Finite Group
11
H. I. Blau Degrees and Diagrams of Integral Table Algebras
19
R. Boltje Canonical Induction Formulae and the Defect of a Character
29
E. C. Dade Counting Characters in Blocks, 2.9
45
H. Ellers The Defect Groups of a Clique
61
K. Erdmann Representations of GL n (K) and Symmetric Groups
67
G. Huang On Extended Block Induction and Brauer's Third Main Theorem
85
R. Kessar On Blocks and Source Algebras for the Double Covers of the Symmetric Groups
93
xii
Table of Contents
L. Puig A Survey on the Local Structure of Morita and Rickard Equivalences between Brauer Blocks
101
G. R. Robinson Some Open Conjectures on Representation Theory
127
L. L. Scott Are All Groups Finite?
133
S. A. Syskin Locally Finite Varieties of Groups and Representations of Finite Groups ,
149
Embeddings, Geometries and Representations: Connections and Computations M. K. Bardoe
Abstract. We outline some of the connections between representations and geometries through the computations of embeddings of geometries. We end with a table of many of the computations which have been completed.
Introduction It is the author's belief that the understanding of groups is simplified through the use of geometries on which these groups act. For instance, the geometries for groups of Lie-type, namely buildings, help make the p-Iocal structure of Lie-type groups over fields of characteristic p easily understood. Also, one may see that of the 26 sporadic simple groups, those which are best understood are those with easily understood geometric structures on which the groups act, e.g. M24, COl, and Fi24. Recently, through the work of Quillen, Brown, and Adem and Milgram, e.g. [AM] and others, it has become clear that understanding the geometries associated to a group may also be helpful in computing and understanding the cohomology of that group at a specific prime. Representation theory is also able to reap the benefits of this viewpoint. What we describe here is a two-way street between representation theory and geometry. One way is the method of constructing from a representation of a group a geometry for that group. This direction is well known and easily understood. The other direction is more complicated. It involves inductively creating modules, based on a geometry associated to a particular group, from modules for its subgroups. We give a short outline of what follows. First, we define geometry and give examples of geometries for some familiar groups. Then we outline some of the theory which describes how to construct representations which are described by geometries. We attempt to explain why this process is interesting to both geometers and to representation theorists. Finally we give a partial list computations and constructions which have been completed.
2
M. K. Bardoe
Geometries We take the following to be as our definition of geometry:
Definition 1: Geometry. A geometry of rank n is an ordered sequence
of n pairwise disjoint non-empty sets fi together with a symmetric incidence relation, *, on their union such that if F is any maximal set of pairwise incident elements then IF n fi I = 1 for each i. These first two examples demonstrate how representations have been used to construct geometries. Example 1: The Projective Geometry for an n -dimensional vector space V, denoted PG(V), is the n - 1 sets given by the I-spaces, 2-spaces, ... , n I-spaces with incidence between elements of different sets defined by inclusion. Example 2: Symplectic Geometry for a vector space V. Let V be a 2n -dimensional vector space with a non-degenerate alternating bilinear form. Then S(V) = (isotropic I-spaces, ... , isotropic n-spaces, *)
with incidence defined as inclusion. This is a geometry defined by the subspaces of a module for the group SP2n(k). This example shows a geometry which comes from a setting other than representation theory. Example 3: The M24 2-local geometry, [RS1]. This geometry is based on the Steiner system 5(5,8,24). In the terminology of our definition f = (Octads, Trios, Sextets, *)
An octad is the special 8-sets which form the blocks of the Steiner system, a trio is a set of mutually disjoint octads, and a sextet is a set of six mutually disjoint 4-sets such that any two 4-sets form an octad. An octad is incident with a trio if it is contained in that trio. An octad is incident with a sextet if it is the union of two of the 4-sets of the sextet. A trio is incident with a sextet if each of its octads is the union of two of the 4-sets of the sextet. In what follows it will be important to be able to view a geometry as a simplicial complex on which the automorphism group acts. This may be done
Embeddings, Geometries and Representations
3
in the following way. Define vertices to be the disjoint union of the fi. Then define n-simplices to be n-sets of V such that any two elements are incident, such a set is called a flag. Such a simplicial complex is of dimension one less than the rank of the geometry, and often goes under the name flag complex. The maximal simplices of this complex are termed chambers.
points (I-spaces) lines (2-spaces)
Figure 1: Geometry for SL 3 (2) viewed as simpIicial complex In the first two examples, PG(V) and S(V), the vertices of the simplicial complex are proper subspaces of V, and the larger dimension simplices are chains of subspaces ordered by inclusion. In the case of PG(V) the chambers are maximal chains of subspaces, while for S(V) chambers are maximal chains of isotropic subspaces.
Sheaf Theory From a group theoretic point of view, geometries are useful ways of describing a portion of the subgroup structure of a group. This may be most evident in the case of buildings and p-Iocal geometries for the sporadic groups. Simplistically, we construct a geometry by taking the conjugacy classes of maximal subgroups containing a Sylow p-subgroup to be the objects of our geometry.
4
M. K. Bardoe
Then say that two such subgroups are incident if their intersection contains a common Sylow p-subgroup. Therefore a natural way to exploit geometries in representation theory is to say we understand something about the way a representation restricts to the stabilizers of objects of a p-Iocal geometry. Then what further can we say about representation for the whole group? This work essentially started, in the case of modular representations, with the work Ronan & Smith [RS2]. Inspired by the work of Lusztig, [L] on the representation theory of Lie-type groups over the complex numbers, Ronan & Smith show how to construct a representation for a group by using the geometry to "weave" representations of the various stabilizers of simplices into a coherent representation for the whole group. This is done through the formalism of sheaves. Let k be a field.
Definition 2: Sheaf. A sheaf J' on f assigns to each simplex a E f a representation J'u of Stab(a) C Aut(r). If r is a face of a there is a linear connecting map cPar: : J'a ---* J'r: such that cPpa 0 cPar: = cPpr: whenever this composite map is defined. The J'a and cPar: are required to be Aut(r)-equivariant in the following sense: For each g E Aut(r) there is a mapping g : J' ---* J' such that gh = gft, J'ag = (J')g, and go cPag,r:g = cPar: 0 g. The homology groups of such a sheaf J' are k Aut(f) -modules. In particular, if we assume that the terms of J' are generated by the images of J'c for chambers, what is called chamber generated, then the zero homology of this sheaf is a module that has the extra condition that one can recover J' through the submodule structure of V ~ Ho(J'). Namely, if a, rare simplices of the simplicial complex derived from f, then there exist submodules Va of V upon restriction Stab(a) such that Va ~ ::1a' Also, if rea, then Va C Vr:' Note that the star of a simplex, St(a), is also a simplicial complex, and that a sheaf, ::1, on f defines a sheaf, 9, for St(a). Therefore if one understands the zero homology of sheaves for St(a) for a of dimension S rank(r) - 2, then one may make the following inductive step: Given a module for the stabilizer of a chamber, Mc, and modules for the stabilizers of the faces of the chambers, Mrr;, with connecting maps, cPc,rri' then one may construct a universal sheaf, U, such that the module at any chamber is isomorphic to Mc and Urr; ~ Mrr ;, and the modules at other simplices are defined to be Ho(St(a)).
Embeddings, Geometries and Representations
Example 4: [R82] Let
5
r
be PG(V) where V is a 3 dimensional space over F2. Then Aut(r) ~ SL3(2). Suppose we define a chamber generated sheaf U by assigning to a chamber a fixed I-space. The face of a chamber associated to a point is assigned a fixed I-space, and the face of a chamber associated to a line is assigned a 2-space which contains the three fixed I-spaces of the chambers which contain the line. In the terminology of our definition of a sheaf, Mp is a trivial module for the point and chamber stabilizers and M[ is the 2-dimensional irreducible for the SL2 (2) quotient inflated to the full tine stabilizer, and the connecting maps are inclusion maps. Then Ronan & Smith show that Ho(U) ~ V for the sheaf defined by these conditions.
Another result of Ronan & Smith, [R82], shows that if one can form a sheaf, ~, from the submodule structure of a module, V, then V is a quotient of Ho(~. Therefore from the last example we see that V is the only module satisfying the condition described by that sheaf, as Ho(U) is an irreducible module for SL3 (2). Therefore this result about the form Ho(~ provides a kind of local recognition result for SL3(2) modules. And in general computation of Ho(~ provides a recognition result for modules of Aut(r). The last example should give you an idea as to how to ask the relevant questions about sheaves but tells you little about how to compute. In general computations are ad hoc in nature and not very enlightening.
Embeddings One area in which many computations have been done has been that of embeddings of geometries. Motivated at least partially by the newer machinery of sheaves and a classical geometric question, a new focus has been centered on the question of what projective geometries can an abstract geometry, such as the one in Example 2, be a subgeometry of. In particular, if we restrict attention to a rank 2 geometry where elements of one set are called points and the elements of the other are termed lines, what projective geometries is this geometry a subgeometry of?
Definition 3: Point-Line Embedding. An embedding of a geometry r is an injective incidence preserving map, rr, from r to PG(V), the projective geometry of a vector space over the appropriate field, such that the points are mapped into the I-spaces of V and the lines are mapped into the 2-spaces of V.
6
M. K. Bardoe
Here is an example of a point-line embedding of the geometry of octads, viewed as points, and trios, viewed as lines, coming from Example 3. Example 5: Geometry of octads and trios from the 2-local geometry for M24. Let V be the 11 dimensional irreducible quotient of the binary Golay code. Then the geometry in Example 2 is isomorphic to
r
= (special I-spaces, special 2-spaces, *)
where incidence defined by inclusion, and special indicates a specific orbit of subspaces under the action of M24. In an attempt to find embeddings for a geometry r, we rephrase this question into the language of sheaf theory in the following way: What modules support the sheaf, ~, given by assigning a I-space to the point stabilizer and a 2-space to the line stabilizer? From what we have said above one such module is Ho(~. This module is known to geometers as the universal embedding of a geometry because there cannot exist a larger embedding of the geometry, and any embedding map is factored by the universal embedding map. The situation is particularly nice when we are working with a geometry with 3 points per line. Notice that if our geometry has three points per line, then the most natural projective space to have our geometry embed into is a projective space coming from a vector space over F2. In the case of F2 vector space we have a unique vector in a I-space. Therefore we can define an embedding as a map n' from r to a set of vectors of V the vector space underlying our projective space. Also, the requirement that the 3 vectors assigned to the points of a line span a 2-space is equivalent to the following equation:
vp
+ vq + v, = 0 for l = {p, q, r}
From this we see that we can write a presentation of the universal embedding by starting first with a vector space, with basis indexed by the points of our geometry, and then quotienting out by the subspace spanned by all of the vectors of the type v p + vq + v, = 0 for l = {p, q, r}.
Conclusion Many universal embedding questions have been determined for many of the simple groups and related groups. These computations may be of interest to geometers because embeddings often are helpful in classification and computational problems. They may be of interest to representation theorists because
Embeddings, Geometries and Representations
7
they show which modules are classified by their restrictions to important subgroups of these groups. Below is a list of many of these results and references for them. In this list irreducible modules are denoted by their dimension, and duality is indicated by a overline. Geometry Building Long root Neimir geometry
Aut(r) Lie-type Group of type A, D or E A7
Near-hexagon
M24
Near-hexagon
U4(3)
Near-octagon
12.2
Near-hexagon
3D4(2)
Near-hexagon
U6(2).2
Near-hexagon
07(2)
Univ. Embedding Natural Module Adjoint Module 0
Remarks [RS2] [SV], [VJ [RS3]
11 IE&TI 20
[RS3]
26 IE&I 26 IE&I 20 IE&I 8 0-
[FS]
T
[Y] [FS]
[Y] [Y]
T
Involution geometry
U4(3)
Involution geometry
Suz
Involution geometry 2-local geometry 2-local geometry 2-local geometry Tilde geometry Tilde'geometry Tilde geometry Tilde geometry Petersen geometry Petersen geometry Petersen geometry Petersen geometry Petersen 'geometry Petersen 'geometry Petersen' geometry Petersen geometry Petersen geometry
Co, Co, He Ru M24 3. Sp4(2) He M Ss Aut(M22) 3. Aut(M22) M23 CO2 3:23. CO2 14
BM 4j/l .BM 3
34 1
EB
-,-
34' 1
[B3]
142
[B2]
274 'E&1E&24
[Bl] [Srn] [MS] [MS] [IS2]
24 51 28 11 6EB5 52 0 6 11 12 EB 11 0 23 23 0 0 0
[IS2] [IS2] [IS2] [IS 1] [IS1] [IS 1] [IS I] [IS3] [Sh] [IS2] [IS2]
8
M. K. Bardoe
Bibliography [AM]
A. Adem and R. 1. Milgram, The cohomology of the Mathieu group M22, Topology 34 (1995),389-410.
[B 1]
M. K. Bardoe, The universal embedding for the Co I involution geometry, in preparation, 1995.
[B2]
M. K. Bardoe, The universal embedding for the Suzuki sporadic simple group, Preprint, accepted to J. Algebra, 1995.
[B3]
M. K. Bardoe, The universal embedding for the U4(3) involution geometry, Preprint, accepted to 1. Algebra, 1995.
[FS]
D. Frohardt and S. Smith, Universal embeddings for the 3 D4(2) hexagon and the h near-octagon, Europ. 1. Combin. 13 (1992), 455-472.
[IS 1]
A. A. Ivanov and S. V. Shpectorov, Geometries for sporadic groups related to the Petersen graph. n, Europ. J. Combin. 10 (1989),347-361.
[IS2]
A. A. Ivanov and S. V. Shpectorov, The flag-transitive tilde and Petersen-type geometries are all Known, BuII. Amer. Math. Soc. 31 (1994),173-184.
[IS3]
A. A. Ivanov and S. V. Shpectorov, Natural representations of the P-geometries of C02-type, J. Algebra 164 (1994), 718-749.
[L]
G. Lusztig, The Discrete Series Representations of the General Linear Groups over a Finite Field, Annals of Mathematics Studies 81, Princeton Univ. Press, Princeton, NJ. 1974.
[MS]
G. Mason and S. Smith, Minimal 2-local geometries for the Held and RudvaIis sporadic Groups, J. Algebra 79 (1982), 286-306.
[RS1]
M. A. Ronan and S. D. Smith, 2-Local geometries for some sporadic groups, in: B. Cooperstein and G. Mason, editors, The Santa Cruz Conference on Finite Groups, Proc. Symp. Pure Math. 37, Amer. Math. Soc., Providence RI, 1980, 283-289.
[RS2]
M. A. Ronan and S. D. Smith, Sheaves on buildings and modular representations of ChevaIley Groups, J. Algebra 96 (1985), 319-346.
[RS3]
M. A. Ronan and S. D. Smith, Computation of 2-modular sheaves and representations for L4(2), A7, 356, and M24, Comm. Algebra 17 (1989),1199-1237.
[Sh]
S. V. Shpectorov, Natural representations of some tilde and petersen type geometries, Geom. Dedicata 54 (1995), 87-102.
[Srn]
Stephen D. Smith, Universality of the 24-dimensional embedding of Comm. Algebra 22 (1995), 5159-5166.
[SV]
COl,
S. Smith and H. Vblklein, A geometric presentation for the adjoint module of 5L3(k), 1. Algebra 127 (1989),127-138.
[V]
H. VOlklein, On the geometry of the adjoint representation of a ChevaIley group, 1. Algebra 127 (1989), 139-154.
Embeddings. Geometries and Representations
[Y]
9
S. Yoshiara, Embeddings of flag-transitive classical locally polar geometries of rank 3, Geom. Dedicata 43 (1992), 121-165.
Imperial College of Science and Technology London SW7 287 England Email:
[email protected] Infinite Dimensional Modules for a Finite Group D. J. Benson
This paper is a transcription of the lecture I gave at the Ohio State University Conference on Representation Theory of Finite Groups. My intention was to talk about the ideas involved in a small corner of my recent joint work with Jon Carlson and Jeremy Rickard [Be2] [Be3] on infinitely generated modules, and try to explain the role of generic points of varieties in this context. As we move into the second century of finite group representation theory, we still find that the vast majority what is being done is concerned with finitely generated modules. It seems to me that the reason for this is largely that we have very few techniques that work in a wider context, say for example the context of arbitrary modules for the group algebra of a finite group over a field. Linear transformations on infinite dimensional spaces don't necessarily have any eigenvalues. There are modules with no indecomposable summands. The Krull-Schmidt theorem fails quite badly, so there are no vertices and sources. There is a module M for Z2 x Z2 in characteristic two, which satisfies M ~ M EBM EBM but not M ~ M EBM. In the light of these pathologies, it is tempting just to give up and return to the relatively safe world of finitely generated modules, especially as there are still many interesting unanswered questions in this context. However, it turns out that even if we are only interested in finitely generated modules, there are recent theorems whose proofs use infinitely generated modules in an essential way; for example, Rickard's (as yet unpublished) classification of the thick subcategories of the stable finitely generated module category for a p-group, and my recent proof [Bel] of the conjectures formulated in [Be4] about finitely generated modules with no cohomology.
1. A Vector Space Lemma The following lemma serves as a replacement for the theory of eigenvalues, and is really the starting point for the recent developments I'm going to discuss. Lemma 1.1. Let k be an algebraically closed field, and k(t) be a simple transcendental extension of k. Let V be a nonzero (and possibly infinite
12
D. J. Benson
dimensional) k-vector space, and let f be linear transformations from V to itself. Thenfor some A in k(t), the linear map
1 0 f - A0 Identity: k(t) 0k V -+ k(t) 0k V is not an isomorphism.
In other words, the reason why there need be no eigenvalues, even over an algebraically closed field, is because the field isn't big enough. Over some extension field, there is always an eigenvalue, if interpreted suitably. The appearance of transcendental extensions in this lemma gives rise to the relevance of "generic points" for infinite dimensional representations, in a way that never becomes relevant for finite dimensional representations. The proof of the lemma is very straightforward. We regard V as a k[x]module in the normal way by letting x act as the linear transformation f. If f - A.Identity is an isomorphism for all A E k, then the action of k[x] extends to an action of the field of fractions k(x). Since k(x) is a field, any nonzero k(x)-module has a summand isomorphic to k(x) itself. But multiplication by 1 0 x - t 0 1 is not an isomorphism on k(t) 0k k(x). The form in which the lemma gets used is the following: if f, g : V -+ W are two maps with the property that after tensoring with k(t), all nontrivial linear combinations of f and g give isomorphisms from k(t) 0k V to k(t) 0k W, then the vector spaces V and Ware both zero. To reduce to the previous form of the lemma, use g to identify V with W.
2. Dade's Lemma The way we use the vector space lemma of the last section is via an infinite dimensional version of Dade's lemma. The original lemma (Dade [Da]) says the following. Let k be a field of characteristic p, and let E = (gl, ... , gr} be an elementary abelian group of order p", Let Xi be the element gi - 1 of the group algebra kE, so that = 0 and Xi is in the Jacobson radical J(kE). Let VE(k) denote the quotient space J(kE)j J 2(kE), and let XI, ... , Xr be the images of X I, ... , X r in VE(k). It is not hard to show that they form a basis for this quotient space. Let YI, ... , Yr be the linear functions VE (k) -+ k given by Yi(Xj) = 1 if i = j and zero otherwise. Then regarding VE(k) as an affine space, its coordinate ring is the polynomial ring k[YI, ... , Yr]. If
Xi
ex = AIXI
+ ... + ArXr E
VE(k)
Infinite Dimensional Modules for a Finite Group
13
is not equal to zero, then we set
an element of order p in the group algebra kE. Theorem 2.1 (Dade's Lemma). Suppose that k is algebraically closed. If M is a finitely generated k E -module such that the restriction of M to (u a ) is free for each point i- a E Vf(k), then M is afree kE -module.
°
The hypothesis that M is finitely generated is certainly necessary here. There are examples of infinitely generated modules for 2/2 x 2/2 which are not projective, but whose restriction to each (u a ) is free (an example is sketched in Section 6). But in some sense, this is because the field is not big enough, because after enlarging the field, one finds that there are values of a for which the restriction to (u a ) is not free. The correct version of Dade's lemma for modules which are not necessarily finitely generated was formulated in [Be3l: Theorem 2.2. Let K be an algebraically closed extension of k of transcendence degreee at least r - 1, and set Vf(K) = 1(KE)/1 2(KE). If M is a kE-module such that (K 0k M) +(u,,} is free for all nonzero a E Vf(K), then M is a free k E -module. The idea of the proof is to reduce to the rank two case, and then choose two linearly independent elements 11, 11' E H 1(E, IFp ) ~ 1(IFpE)/1
so that the Bocksteins f3(11) and of H 2 (E, IFp). They induce maps
2(IF
pE),
f3(11') are algebraically independent elements
with the property that for all A, f-LE K, not both zero, Af
+ ug
~
---2
: ExtKE(K, K 0k M) -+ ExtKE(K, K 0k M)
is an isomorphism. It now follows from the vector space lemma discussed earlier, that Ext~E(k, M) = 0, which implies that M is free.
14
D. J. Benson
3. The Rank Variety For M a finitely generated kE-module, with k algebraically closed, Carlson's definition [Ca] of the rank variety of M is VE(M)
= {OI- ex E
VE(k)
IM
,l.(u a )
is not free} U {O}.
This is a closed homogeneous subvariety of VE(k), and Dade's lemma may be interpreted as saying that M is free if and only if VE(M) = {O}. In fact, the dimension of VE(M) determines the polynomial rate of growth ofthe minimal free resolution of M as a k E -module, which is called the complexity of the module. Thus for example the dimension of VE(M) is equal to one if and only if the minimal resolution of M is periodic (after the first term, which may be too big because of the free summands of M). For infinitely generated modules, the theory is somewhat different, because we must extend the field in order for Dade's lemma to hold. The naive thing to do is just to look at VE(K ®k M), where K is a "large enough" transcendental field extension of k. In general this is not a closed subset of VE(K), so what sorts of subsets occur this way? To answer this question, we next discuss the theory of generic points.
4. Generic Points For this section, we suppose that k is algebraically closed, and that K is an extension of k of transcendence degree at least r. Recall that if V s:;VE(k) is a closed irreducible subvariety, then the set p = {f
E
k[YI, ... , Yr]
I f vanishes on
V}
is a prime ideal in k[YI, ... , Yr], and k[V] = k[YI, . " , Yr]/p
is the coordinate ring of V. It is an integral domain, and its field of fractions key) is the function field of V. It is generated as an extension field of k by the images YI,... ,Yr of the elements YI, ... , Yr. The transcendence degree of key) over k is equal to the Krull dimension of k[V], and is by definition the dimension of the variety V. Since K is an algebraically closed extension field of k of transcendence degree at least as big as that of key), it follows that the inclusion of k into K extends (not by any means uniquely) to an embedding of key) into K. Let
Infinite Dimensional Modules for a Finite Group
15
tl, ... , t, be the images of YI, ... , Yr under such an embedding. The generic point of V is defined to be the element YIXI
+ ... + Yrxr
E
VE(k(V».
Note that this point is well defined, independently of the chosen basis for VE(k). A point which is of the form tlxl
+ ... + t.x,
E
VE(K)
for some embedding of k(V) into K as above, is said to be a generic point of V over K. If tlxl + ... + t.x, is any point in VE(K), set p S; k[YI, , Yr] equal to the ideal consisting of all polynomial relations satisfied by ti, .t, over k. Since K is a field, this ideal is necessarily prime. Let V be the associated subvariety of VE(k). Then the point tlxl + ... + t-x, is a generic point of V over K. So every point defined over K is generic for some uniquely determined closed irreducible subvariety defined over k. If we look at a line through the origin in VE(K), the points in that line may be generic for possibly different subvarieties. However, there is a uniquely determined homogeneous subvariety (i.e., one which is a union of straight lines through the origin) among them. To see this, if tlXI + ... + t.x, is generic for some inhomogeneous subvariety V, then the dimension of V is less than r, so there is an element A E K which is algebraically independent of tl, ... , t.. Then the point Mlxl + ... + Mrx r is generic for the homogeneous hull of V, namely the smallest homogeneous subvariety containing V. Now if M is a kE -module (not necessarily finitely generated), then the question of whether (K 0k M) -J-.(u a ) is free only depends on the line through a in VE(K), and then only on the closed homogeneous irreducible subvariety of VE(k) for which it is generic. So we define VE(M) to be the collection of nonzero closed homogeneous irreducible subvarieties V of VE(k) with the property that if a is the generic point of V then (k(V) @k M) -J-.(u a) is not free. It is easy to see that each of VE(K 0k M) and VE(M) determines the other.
5. Properties of
VB(M)
The following list of properties of VE(M) when M is not necessarily finitely generated parallels the list of properties of VE(M) in the finitely generated case. (i)
VE(M) = 0 if and only if M is projective.
16
D. J. Benson
(ii) More generally, the dimension of V is at most e for all V E V'E(M) if and only if M may be expressed as a filtered colimit of finitely generated
modules of complexity at most e. (iii) V'E(M EBN) = V'E(M) U V'E(N). (iv) V'E(M 0k N) = V'E(M) n V'E(N). (v) Every subset of V'E(k) = Projk[YI, ... , Yr] is equal to V'E(M) for a suitable module M. (vi) If M happens to be finitely generated, then V'E(M) is just the collection of all closed homogeneous irreducible subvarieties of VE(M).
6. An Example Let k be an algebraically closed field of characteristic two, and let E = Zj2 x Zj2 = (gl' g2). Set XI = g, - 1 and X2 = g2 - I. Let M, be the kE-module with generators ml, m i, ... and relations Xlml = 0, and X2mi = X,mi+1 for each i ~ 1. Let M2 be the kE-modulewithgenerators m;, m;, ... and relations X2m; = X I m;+1 for each i ~ 1. The modules M, and M2 are superficially similar, but MI has complexity one, while M2 has complexity two. The set V'E(MI) consists of just a single line through the origin, while
V'E(M2) = V'E(k) \ VE(M,). Let 0 (iff SUPPB(bb) = {I}) [AI]. If b = C E Cla(G) then b is linear iff C S; Z(G) iff ICI = 1. If b = X E Irr(G), then b is linear iff X(l) = 1. There are several useful ways to construct new table algebras from old. One of them is called rescaling [A 1, Bll]: given table algebra (A, B), choose AI, A2, ... , Ak E lR.+, subject only to AI = 1 and AI = Ai for all i. Let B' = {Aibi I bi E B}. It is easily seen that (A, B') is another table algebra, with respect to the same automorphism - and homomorphism f. _
A
Degrees and Diagrams of Integral Table Algebras
21
Another easy construction is symmetrization [A2, Section 3], the prototype for which comes from commutative association schemes [BI, p.57]. For all bi E B, define b? := bi if bi is real, b? = br := bi + bi if not. Let BD := {b? I bi E B} and A- := {a E A I a = a}. Then (A-,B o), the symmetrization of (A, B), is a table algebra with all elements real, and is integral if (A, B) is. Let (A, B) be a table algebra with B = {bl = 1, b2, .. " bd. Fix bE B. Then for 1 :::: i :::: k, k
bbi = Ldjibj , j=1
for unique dji
E
lR.+ U {O}.
Definition. [BI2] (See also [BI, p. 114].) The representation graph of B with respect to b (denoted rh (B) ) is the directed graph with vertex set {I, 2, ... , k} (in bijection with B), and where there is a directed edge from vertex i to j (which is labeled by dji ) iff dji > O. It is clear that rb(B) is connected if and only if b is faithful. If b is real, then dij > 0 {} dji > 0, and so rb(B) may be presented as an undirected, labeled graph, where each pair of adjacent vertices and corresponding edges appears as j
and whenever dii > 0, there is a loop:
;0
(d;.).
Suppose that (A, B) is an integral table algebra. Ifthere exists b E B which is faithful, real and has degree f(b) = 2, then rb(B) is the underlying graph of a generalized Cartan matrix (as in [HPR]), on whose index set {I, 2, ... , k} there is an additive function given by the values of f on B [BI2, Proposition 5.8]. The generalized Cartan matrices with an additive function are classified by Vinberg (as in [HPR]), in terms of their graphs. Twenty such graphs exist, 11 of which represent infinite families, and they are called the generaLized EucLidean (or affine) diagrams. A study of the diagrams leads to the following.
22
Harvey I. Blau
Theorem 1 [BI2, Theorem 2]. Exactly 13 of the 20 generalized Euclidean diagrams occur as representation graphs of integral table algebras with a faithful real basis element o[ deEf.ree~. T~e 7Aagr~s which ~o not occur are (in the notation of [HPRJ) An, AI2, B n , L n , BL n , CD n and G2I. Each of the 13 which do occur determines an integral table algebra to exact isomorphism, with the single exception of Dn, for which there are precisely two exact isomorphism classes of integral table algebras.
Remarks. (1) A slightly broader method of assigning labeled graphs to integral table algebras as in Theorem 1 produces 19 of the 20 diagrams, and thereby generalizes the realization of the affine diagrams from the finite subgroups of SL(2, C), as in [M, SI] (see [BI2, Theorem 3].) (2) The integral table algebras with a faithful nonreal element of degree 2 also are classified, under the additional assumption that there are no nontrivial linear elements of degree 2m , for any m :::: 0 [BI2, Theorem I]. We omit here the specific details of the conclusion of Theorem 1, but two examples may be illuminating.
Example 1. Let G be the dihedral group of order 2(2n
+ 1),
n > O. Let B be the subset of Cla(G) which consists of those class sums C such that C S; (x) = Z2n+I, the~nique subgroup of index 2. Let A = (B), b = x + x-I. Then r b (B) is CL n, where all nonzero dj i = I except as noted, and all degrees are as listed:
~1_(_2_'1_)
:
2
~
;
for G = SL(2, 5). If b is an irreducible character of degree 2, then rb(B) is £8, with all nonzero dji = I and all degrees as listed:
Example 2. Let (A, B)
= (Ch(G), Irr(G»
3
]
b 2
3
4
5
6
4
2
Degrees and Diagrams of Integral Table Algebras
23
Question. What can be said about integral table algebras and their diagrams when all faithful elements have degrees larger than 2? No analog of the Vinberg classification seems known in this much more general context, and the complexity ofthe problem, even at degree 3, increases enormously. In order to gain some insight into possible approaches, we have been studying integral table algebras where all nontrivial basis elements have degree 3. Definition. [BXI] A table algebra (A, B) is called homogeneous (of degree A) iff IBI > I and, for some fixed A E ~+, f(b) = A for all b E B\ {l}. Example I, of course, is homogeneous of degree 2. It is not all that restrictive to assume that an integral table algebra is homogeneous of some particular degree, in view of the following observation. Theorem 2 [BX1]. Any integral table algebra has a rescaling which is also integral, and which is homogeneous of some positive integer degree A. This means that a complete classification of homogeneous integral table algebras is rather a tall order. The degree A which results from rescaling an arbitrary integral table algebra, as in Theorem 2, is usually very large. Example 3. [BXI] Fix A E ~ with A > 1. Let a = (A - 1)/2, f3 = (A + 1)/2. For each integer m ~ 0, we define an integral table algebra (A, Tm(A). Let Tm(A):= {l,xO,XI, ... ,Xm } be a basis for an (m+2)dimensional vector space A over C, and define products of these vectors so that I is the multiplicative identity, and with, for i, j ::: m,
°::
XiX) =
I
aXi+j+{3Xi+J+I A·I + axO + (U m aXi+)-m + {3Xi+)-m-1
if if if
i + j < m, i + j = m, i + j > m.
Then (A, T m (A)) is a table algebra, with Xi = Xm-i for all i, and which is homogeneous of degree A. If A ~ 3 is an odd integer, then (A, Tm(A)) is integral. If m is even, then X m j2 is real and faithful, and the representation graph lXm /2 (Tm (A)) is
24
Harvey I. Blau
(13./1) X
(;\
m
(a,a)
(/1,13)
(0.0) Xm./2_1
Im-l
........~
X m -2
X m /2_2
~'
(A,I) X m /2
(Q 'Q)
x,
I m /2+1
(a,a)
({3,{3)
(a.n)
(13.13) Xo
x,
X m !1+2
-- - - ----
We consider briefly the symmetrization (A -, T m(A)O). Order the basis as
{I, Xm/2, XO + Xm , Xm/2-1 + Xm/2+1, XI + Xm-I, Xm/2-2 + Xm/2+2, ... } . The multiplication of these elements by matrix of structure constants
a
1
A a a 2a a a a a a a
a a a a a {3 {3 a a a a a
where 8 = a or {3, as m
X m /2 =
X~/2 yields the tridiagonal
a a a a a a a a a a a a a a a a a {3 a a {3 a a a a a {3
== a or
8
a
a
A-8
A- 8
8
2 (mod 4). Then the graph rxo (Tm(A)O) m(2
is
(>.,1)
(l3,il)
(2a,a)
A
2A
(a.o) 2A
(3. J)
(a,a)
(J,J) 2,\
2A
(A - J,'\ - J) 2A
(J)
2,\
Thus, (A - , T m (A)O), for m even, is a table algebra of P -polynomial type [BI, p. 317], that is, each element of Tm(A)O is a polynomial in Xm/2.
Degrees and Diagrams of Integral Table Algebras
25
The T m (A)Q do not arise from association schemes, as in [Br, Section IILl], since the sequences of subdiagonal entries and of superdiagonal entries are not monotonic (see [Br, Proposition III.I.2l). The existence ofthese P -polynomial table algebras was also observed directly, and independently, by Arad and Muzychuk. The question of which tridiagonal matrices with given nonnegative integer entries yield integral table algebras of P -polynomial type seems highly nontrivial. Example 4. Let H be an abelian group which admits a fixed-point-free action by Zn (cyclic of order n), for some n > O. Let G be the semi-direct product HZ n , let B consist of all C E Cla( G) such that C ~ H, and set A = (B). Then (A, B) is an integral table algebra which is homogeneous of degree n. (Example I is clearly a special case of Example 4.) Remark. The table algebras Tm(A) for m ~ 2 never arise as (Z(CG), Cla(G)), (Ch(G), Irr(G)), or any substructure thereof, for any finite group G. The equation XQXm-1 = CUm-1 + f3iQ holds in T m (A), whereas there existnoelements b, c in Cla(G) or Irr(G) with c =I- b, b =I- b, and SUPPB(bc) = b} [AI, Corollary E'].
rc,
We recently have determined all integral table algebras (A, B) which are homogeneous of degree 3, and for which B contains a faithful, real element [BX2]. The full result is too detailed to give here, so we restrict to the case where B contains no nontriviallinear elements. First, we need to introduce a few more table algebras. Example 5. Three table algebras, with table bases V2, V3, fined by the following products of nontrivial basis elements. := {I, v}, where v 2 = 6·1
V3
:= {I, VI, V2}, where V[ = 3·1
{I, VI,
V2, V3},
resp., are de-
+ v;
V2
V4:=
V4
+ 2V2; VI V2 = V2VI = 2vI + V2, vi = 3· 1 + vI + V2; where V[ = 3·1 + VI + V2 = v5, vi = 3·1 + 2V2, VI V2 = V2VI = VI + 2V3, VI V3 = V3VI = 2V2 + V3, V2V3 = V3V2 = 2vI + V3.
It is easy to check that V2, V3, V4 are indeed table bases for integral table algebras (with trivial automorphism) which are homogeneous of degree 3.
26
Harvey I. Blau
Theorem 3 [BX2]. Let (A, B) be an integral table algebra which is homogeneous of degree 3, and such that B contains afaithful real element and no nontriviallinear elements. Then B is exactly isomorphic to one of V2, V3, V4 or T m (3) for some even integer m :::: o. Arad, Fisman, Muzychuk and Miloslavsky, in recent work [AFM], have obtained some results on homogeneous integral table algebras of degree 3 which have no nontrivial real elements in the table basis. Many instances of Example 4 are of this type, as is Tm (3) for m odd. The main theorem of [AFM] characterizes the algebras of Example 4 when n = 3 and IH I is odd.
References [AI]
Z. Arad and H. L Blau, On table algebras and applications to finite group theory, J. Algebra 138 (1991),137-185.
[A2]
Z. Arad, H. I. Blau, 1. Erez and E. Fisman, Real table algebras and applications to finite groups of extended Camina-Frobenius type, J. Algebra 168 (1994), 615-647. Z. Arad and E. Fisman, On table algebras, C -algebras and applications to finite group theory, Comm. Algebra 19 (1991), 2955-3009.
[AF]
[AFM] Z. Arad, E. Fisman, V. Miloslavsky and M. Muzychuk, On anti symmetric homogeneous integral table algebras of degree 3, Bar-Ilan University preprint. [BI] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjaminl Cummings, Menlo Park, 1984. [Bll] [BI2]
H. L Blau, Quotient structures in C -algebras, J. Algebra 175 (1995), 24-64. H. L Blau, Integral table algebras, affine diagrams and the analysis of degree two, 1. Algebra 178 (1995), 872-918.
(BX 1] H. I. Blau and B. Xu, On homogeneous integral algebras, 1. Algebra, to appear. [BX2] H. L Blau and B. Xu, Homogeneous integral table algebras of degree 3, Northern Illinois University preprint. R. Brauer, On pseudo groups, 1. Math. Soc. Japan 20 (1968), 13-22. [Br] [HPR] D. Happel, U. Preiser and C. M. Ringel, Binary polyhedral groups and Euclidean diagrams, Manuscripta Math. 31 (1980),317-329. [K] Y. Kawada, Uber den Dualitatssatz der Charaktere nichtcommutativer Gruppen, Proc. Phys. Math. Soc. Japan (3) 24 (1942), 97-109. [McK] 1. McKay, Graphs, singularities and finite groups, in: The Santa Cruz Conference on Finite Groups (B. Cooperstein, G. Mason, eds.), Proc. Sympos. Pure Math. 37, Amer. Math. Soc., Providence, R.L, 1980, 183-186. [McM] R. McMullen, An algebraic theory of hypergroups, Bull. Austral. Math. Soc. 20 (1979), 35-55.
Degrees and Diagrams of Integral Table Algebras
27
[Sc]
I. Schur, Zur Theorie der einfach transitiven Permutations-gruppen, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. K1, 1933,598-623.
[SI]
p. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Math. 815, Springer-Verlag, Berlin, 1980.
Department of Mathematical Sciences Northern Illinois University DeKalb, IL 60115 U.SA EmaiJ:
[email protected] Canonical Induction Formulae and the Defect of a Character R. Bo/tje
Abstract. We explain the idea and the machinery of canonical induction formulae with going as little into details as possible and show that a certain version keeps track of the i-defect d(X) of an irreducible character X of a finite group G by mapping X precisely to the d (X)-th layer in the filtration A ~ i-I A ~ i- 2 A ... , for a prime i and a certain free abelian group A.
1. Introduction and Preliminaries Since its publication in 1947 Brauer's induction theorem (see [Br47]) which says that each character of a finite group is expressible as an integral linear combination of induced one-dimensional characters of elementary subgroups has become a cornerstone both in group theory and number theory. The idea that there are distinguished ways of expressing a character according to Brauer's theorem (at least if one allows all subgroups instead of only elementary subgroups to induce from) originated independently with V. Snaith, cf. [S088], and the author, cf. [B089]. To explain what this means we have to introduce some notation.
1.1.
For a finite group G let R(G) denote the character ring of G and R?:.o(G) the additive monoid of characters of G, i.e. R(G) is the free
abelian group on the set Irr(G) of absolutely irreducible characters of G and R?:.o(G) S; R(G) is the set of all non-negative integral linear combinations of elements in Irr(G). Furthermore, let Rt(G) be the free abelian group on G-conjugacy classes [H, lp]e of the poset M(G) of pairs (H, lp) where H is a subgroup of G and lp is a one-dimensional character of H. The poset structure on M(G) is defined by (K, 0/) ::::: (H, lp), if and only if K .:::: Hand 0/ = res (lp), and it is respected by G -conjugation. The group R~(G) has already been introduced by Deligne in [De]. Using methods from topology, Snaith defined in [S088] a function
f
fe: R?:.o(G) ~ R~(G)
30
R. Boltje
which is natural with respect to restrictions. In fact, for H ::: G there is a restriction map
res+~: R~(G) ---+ R~(H),
[K, 1/I]c
f-+
L
[H n gK,
res~~gK (g1/l)]H,
gEH\C/K
where the sum runs over representatives g E G of the double cosets H\ G / K , gK := gKg- 1 and g1/l(gxg- 1) := 1/I(x) for x E K. This restriction map has been defined by Deligne (cf. [De]) and can also be naturally interpreted by viewing R~ (G) as the Grothendieck group of a suitable category of monomial G-modules, cf. [B095, Chapter 5]. Snaith's map lC is a canonical induction formula, where 'canonical' means that lc commutes with restrictions, and 'induction formula' means that b c 0 lC = idR(C), where bc is defined as bc: R~(G) ---+ R(G),
[H, of its normal subgroup N G (C). We denote by NE (C, ef» the stabilizerin NE (C) of any such ef>, and by N E(C, ef» = s(NECC, ef») the image of that stabilizer in E. Since N G (C) is contained in NE (C, ef», the exact sequence (2.2) restricts to an exact sequence
1 -+ NG(C)
E Irr (NG (C) ) . In addition to the p-block B of G and the integer d :: 0, we now fix a subgroup F of E. Definition 2.4. For any p-chain C of G we denote by k(C, B, d, number of characters ef> E Irr(NG (C)) satisfying d(ef» = d,
B(ef»
G
= Band
N E(C, ef»
F) the
= F.
It is straightforward to verify that the integer k( C, B, d, F) depends only on the G-conjugacy class of the p-chain C. So the sum in the following conjecture is well defined. The Invariant Conjecture 2.5. If Op(G) = 1 and d(B) > 0, then '" L
(-1) ICI k(C,B,d,F)=O.
(2.6)
CE':R(Gl/G
The above invariant conjecture is the equivalent in our situation of the form of Alperin's Weight Conjecture considered by Robinson and Staszewski in [RS]. It reduces to the Ordinary Conjecture 1.4 when E = G. Furthermore, it implies that conjecture in any case, since the equation (1.5) can be obtained by summing the equation (2.6) over all subgroups F of E.
Counting Characters in Blocks, 2.9
49
3. The Extended Conjecture The Invariant Conjecture 2.5 is reasonably easy to verify or refute for any given group G. But it is not strong enough for an inductive proof. To carry out such a proof we need to calculate, for each p-chain C of G, the number of irreducible J-characters 1/1 of NE(C) with a given defect lying over a fixed irreducible J-character q; of Nc(C). Except in rare cases (which, in fact, are not so rare for simple G), the subgroup N E( C, q;) of E by itself does not determine this number. However, the Clifford extension for q; does determine it. So we reformulate the conjecture in terms of Clifford extensions. We start from the exact sequence (2.2) for a p-chain C of G. To each q; E Irr(Nc(C)) is associated a central extension E[C,q;,J] of the unit group U (J) of the field J by the stabilizer NE (C, q;) of q; in NE (C). Thus we have a new exact sequence O. then '"' ~
- a) = O. (-1) ICI k(C, B, d, F,
(3.4)
CEX(C)/C
This conjecture implies the Invariant Conjecture 2.5 since the equation (2.6) can be obtained by summing (3.4) over all a E H2 (ft, U(J»). If we know that the exact sequence (3.1) must split for all p-chains C of G and all irreducible J-characters c/J of Nc(C), then the Extended Conjecture 3.3 is equivalent to the Invariant Conjecture 2.5. This is notably the case when the group E ~ E / G is cyclic (see [03, 11.20 and 6.5]). It even occurs when the Sylow r -subgroups of E are cyclic for each prime r.
4. Projective Conjectures The next problem to face in trying to construct an inductive proof of the conjecture is that Clifford theory starts from irreducible ordinary characters but ends with irreducible projective ones. Thus it leads us from one situation to a different one, to which we cannot apply the Extended Conjecture 3.3. Howevever, Clifford theory works just as well for projective characters as it does for ordinary ones, and it always gives back projective characters. So we must reformulate our conjectures in terms of projective characters if we wish to prove them inductively using Clifford theory. We begin by replacing the finite group E by some central extension E[J] of U (J) by E. So we fix an exact sequence 1 ---+ U (J)
~
E[J]
~E
---+ 1
(4.1)
of groups such that U (J) is a central subgroup of E[J]. If H is any subgroup of E, then H[J] will denote the inverse image of H in E[J]. SO H[J] is a
Counting Characters in Blocks, 2.9
51
central extension of U (~) by H. We shall assume that ~ is a total splitting field for the extension E[~], in the sense that every irreducible projective ~-character 1J of the subextension H[~] is absolutely irreducible for every subgroup H of E. Then the degree 1J (1) of any such 1J divides the order IH I of the finite group H. So 1J has a defect d (1J), the largest integer d ~ 0 such that pd divides the integer IH I/1J (1). Because the group E[~] is totally split as an extension of U(~), it has a unique subgroup E[91] intersecting U(~) in U(91) and having E as its image (see [D3, 7.8]). Thus the restriction of l][~] is the epimorphism l][91] in an exact sequence I
~ U(91)~ E[91]~
E
~ 1
(4.2)
making E[91] a central extension of U(91) by E. The inverse image H[91] in E[91] of any subgroup H of E plays the same role for H[~] as E[91] plays for E[~]. The twisted group order D[H] of Hover 91 corresponding to the extension H[91] is a suborder spanning the twisted group algebra 2l[H] of H over ~ corresponding to the extension H[~]. So we may define the projective p-blocks of H[~] to be the blocks of this unique 91-order D[H]. We denote by Blk(H[91]) the set of all projective p-blocks of H[~]. Any irreducible projective ~-character 1J of H[~] is an irreducible character of 2l[H], and hence belongs to a unique projective p-block B(1J) of H[~]. The natural conjugation action of H on its twisted group algebra D[H] over 91 has the center Z(D[H]) as its set of fixed points. So we can use this action to define defect groups D :s H for any block b E Blk(H[91]) (see [D3, 9.3]). The defect d(b) of b is then the non-negative integer such that pdCb) is the order of each such D. Brauer induction of projective blocks can be defined just as was Brauer induction of ordinary blocks (see [D3, 10.5]). The equivalent of [KR, 3.2] holds for projective blocks by [D3, 10.14]. Once we change the misprinted "2l[NG(P)] " in the latter proposition to the correct "2l[NG(C)]," it tells us that any projective p-block b of NG (C)[~], for any p-chain C of G, induces a projective p-block bG[J] of G[~]. As usual, we fix a non-negative integer d. The earlier p-block B of G is now replaced by a fixed projective p-block B[~] of the central extension G[~] of U (~) by G. The projective equivalent of Definition 1.3 is Definition 4.3. For any p-chain C of G we define k(C, B[~], d) to be the number of irreducible projective ~ -characters 1J of N G (C) [~] such that d(1J)
=d
and
B(1J)G[J]
= B[~].
52
E.C.Dade
The resulting number k(C, B[J], d) depends only on the G-conjugacy class of C. The projective equivalent of the Ordinary Conjecture 1.4 is
The Projective Conjecture 4.4. If Op(G) = I and d(B[J]) > 0, then
L (- I),c'k(C, B[J], d) = O.
(4.5)
CE::R(GJ/G
This is the conjecture given in [D3, 15.5], but written in a slightly different form. So it implies the Alperin-McKay Conjecture by [D3, 17. IS and 18.5]. Of course it reduces to the Ordinary Conjecture 1.4 when G[J] is a split extension of U (J) by G. If C is any p-chain of G, then its normalizer N E(C) acts naturally by conjugation on NE(C)[JJ, centralizing the subgroup U(J). Since NG(C) is a normal subgroup of NECC), this action leaves NG(C)[J] invariant, and hence permutes among themselves the characters 4> E Irr(NG(C)[JJ). We denote by NE (C, 4» the stabilizer in NE (C) of any such 4>, and by Ni (C, 4» the image Ni(C,
of that stabilizer in
4» =
e(NE(C,
4»)
E. SO C and 4> determine an exact sequence
just as in (2.3). In addition to the above d and B[J] we now fix a subgroup Then the projective equivalent of Definition 2.4 is
F
of
E.
Definition 4.6. For any p-chain C of G we denote by k( C, B[J], d, F) the number of irreducible projective J-characters 4> of NG(C)[J] satisfying d(4)) = d,
B(4))G['5J = B[J]
and
Ni(C,
4» = F.
The number k(C, B[J], d, F) depends only on the G-conjugacy class of C. The projective equivalent of the Invariant Conjecture 2.5 is
The Invariant Projective Conjecture 4.7. If Op(G) = I and d(B[J]) > 0, then
L
(_l)ICik(C, B[J], d,
F) =
O.
(4.8)
CE::R(GJ/G
Of course this reduces to the Projective Conjecture 4.4 when E = G, and to the ordinary Invariant Conjecture 2.5 when E[J] is a split extension of U(J) by E.
Counting Characters in Blocks, 2.9
53
It is well known that Clifford theory works just as well for projective characters as it does for ordinary ones (see [D3, 12.12]). In particular, associated with each p-chain C of G and each character ep E Irr( NG(C)[~]) is a central extension E[C, ep,~] of U(~) by N E(C, ep). Our assumption that E[~] is totally split over ~ implies that E[C, ep,~] is also totally split over ~ (see [D3, 11.20]). There is a one to one correspondence between all characters 1/1 E Irr( N E(C)[~J) lying over ep and all characters 1/1' E Irr( E[C, ep, ~]). Furthermore, the number of such 1/1 with a given defect can be computed from the defect of ep and the cohomology class a [C, ep, ~] E H 2 ( NE (C , ep), U (~) ) corresponding to the extension E[C, ep, ~]. In addition to the above d, B [~] and F, we now fix an element a in the cohomology group H 2 (F, u (~»). The projective equivalent of Definition 3.2 is
Definition 4.9. For any p-chain C of G we denote by k(C, B[~], d, F, a) the number of irreducible projective ~-characters ep of NG(C)[~] such that
deep) = d,
B(cP)G[J]
=
B[~],
N E(C,
ep) = F
and
a[C, cP,~]
= a.
The number k(C, B[~], d, F, a) also depends only on the G-conjugacy class of C. The projective equivalent of the Extended Conjecture 3.3 is The Extended Projective Conjecture 4.10. If Op(G) = I and then 'L"
- a) = O. (-1) ICI k(C, B[~], d, F,
d(B[~])
> 0,
(4.11 )
CE~(G)/G
This reduces to the ordinary Extended Conjecture 3.3 when E[~] is a split extension of U (~) by E. It also implies the Invariant Projective Conjecture 4.7, and reduces to that conjecture whenever each a[C, ep,~] is known to be trivial for all p-chains C in G and all ep E Irr( NG(C)[~J). This last situation occurs in the case where E :::::= E / G has cyclic r -Sylow subgroups for all primes r.
5. The Inductive Conjecture We have still not reached the final modification needed to state a suitable inductive form of the conjecture. The remaining problem can be understood by considering our usual p-chain C of G and character ep E Irr( NG(C)[~J). The associated Clifford extension E[C, cP,~] does determine the degrees of
54
E.C.Dade
the characters 1/1 in Irr( NE(C)(J]) lying over cP, but it does not tell us to which projective p-block of NE( C) [J] any such 1/1 belongs. We need that extra information in order to compute the projective p-block of E[J] induced by B( 1/1). To obtain it we invoke the Clifford theory for blocks developed in [Dl]. Let b be any projective p-block of NG(C)[J]. We denote by NE(C, b) the stabilizer of b under conjugation by elements of NE (C), and by Ni (C, b) the image N i(C, b) = e(NE(C, b))
of that stabilizer in E. In the language of [Dl, 2.17] our present Ni(C, b) would be called Ni(C)h. We denote by Ci(C, b) the normal subgroup of N i(C, b) which would be called N i(C)[b] in the language of[Dl, §2]. This notation is chosen because Jacobinski has noticed that the inverse image CdC, b) = e- J (Ci(C, b))
n NE(C)
of C i (C, b) is precisely the subgroup of all a E NE (C) which centralize to within inner automorphisms the indecomposable direct summand of the 9t-order D[NG(C)] corresponding to b. The Clifford extension for the block b and the exact sequence (2.2) is a central extension E[C, b,~] of U(~) by C i(C, b). So it appears in an exact sequence _
I
-7
<J
_
U (J) ~ E [ C, b, J]
1)[C,h.JJ
) C i (C , b)
-7
1
(5.1)
of groups in which U(~) is a central subgroup of E[C, b, ~]. In the language of[Dl, 2.13] the extension E[C, b,~] would be called N i(C)[b]*. There is a natural conjugation action of the group Ni (C, b) as automorphisms of the central extension E[C, b,~] (see [Dl, 2.19]). Clifford theory for blocks [Dl, 3.7] gives us a one to one correspondence between all projective p-blocks fJ of N E(C, b)[J] lying over b and all N i(c' b)-conjugacy classes of projective blocks fJ' of the central extension E[ C, b,~] of U (~) by C i(C, b), Le., of all blocks fJ' of the twisted group algebra of C i(C, b) over J defined by that extension. Assume that the above block b induces the projective p-block B[J] of G[J]. The latter block has a normalizer Ni(B[J]) andacentralizer Ci(B[J]) in E. It also has a Clifford extension E[B[J]'~] of U(~) by Ci(B[J]), and a conjugation action of N £(B[J]) as automorphisms of that Clifford extension. Since b induces B[J], its normalizer N i(C, b) is a subgroup
Counting Characters in Blocks, 2.9
55
of NE (B [J]). A simple extension of the arguments in [DI, 8.1] shows that CE(B[J]) is a subgroup of CE(C, b), and hence is a normal subgroup of N E(C, b). Furthermore [DI, 8.1] also tells us that the Brauer homomorphism Brc = Br p associated with the final p-subgroup P = Pn in C sends E[B[J],~] monomorphically into E[C, b,~] in such a way that the following diagram commutes U(J)
E[B[J],
1~
l
U(J)
J]
I][B[J]'~l )
src
E[C, b, J]
I][C,b.~J )
C E(B[J])
~
1~ CE(C, b)
(5.2) ~
and the conjugation actions of N E(C, b) on E[B[J],~] and E[C, b,~] are preserved. We have already remarked that J is a total splitting field for the Clifford extension E[C, cP, J] of U(J) by N E(C, cP) associated with any irreducible projective J-character cP of Nc(C)[J]. As in (4.2), this implies the existence of a unique subgroup E[C, cP, 91] of E[C, cP, J] covering N E(C, cP) and intersecting U(J) in U(91). Factoring E[C, cP, 91] by the kernel 1 + P of the natural epimorphism of U(91) onto U(J), we obtain a central extension E[C, cP,~] of U(J) by N E(C, cP). We call E[C, cP, J] the residual Clifford extension for the character cjJ. Suppose that the above character cP lies in the projective p-block b of NcCC)[J]. Then N E(C, cjJ) is a subgroup of N E(C, b). Furthermore, [DI, 13.61 tells us that NE(C, cP) contains CE(C, b). By [DI, 13.101 there is a natural monomorphism /l[C, cP] of E[C, b, Jl into E[C, cP, J] such that the following diagram commutes U(J)
E[C, b,
1=
1
U(J)
E[C,
J]
I][C,b,JJ )
(5.3)
JLlC '4>J
cP, J]
I][C,ep,JJ )
and the conjugation actions of N E(C, cP) on E[C, b, Jl and on E[C, cP, J] are preserved. When the block b containing cP induces B[J], we may compose the monomorphisms in the commutative diagrams (5.2) and (5.3) to obtain a mono-
56
E. C. Dade
morphism p,[B[J], C, cP] = jL[C, cP]
0
Brc: E[B[J], J]
>---;
E[C, cP, J]
of groups such that the following diagram commutes
1 -----+
ry[B[JJ,JJ
], c,
1= 1 -----+ U(J)
E[B[J], J] JL [
]
Ci(B[J)) -----+ 1
1~
(5.4)
Ni(C,cP)
-----+ 1
ry[C,c/J,JJ
s(q) = (l - q)
i=l
(l _
q(p-di)pfi) HI'
(l-qP'
)
This is proved by using a contour integral. For details we refer to [E2].
References [B]
D. Benson, Modular representation theory: new trends and methods, Lecture Notes in Math. 1081, Springer-Verlag, 1981.
[CL]
R. Carter, G. Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193-242.
[CPS]
E. Cline, B. Parshall, L. L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988),85-99.
[CPS2]
E. Cline, B. Parshall, L. L. Scott, Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 59 I, Vol. 124, 1996.
[CP]
C. de Concini, C. Procesi, A characteristic-free approach to invariant theory, Adv. Math. 21 (1976),330-354.
[DR]
Y. Dlab, C. M. Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 (1989), 280-291.
[D I]
S. Donkin, Rational representations of algebraic groups, Lecture Notes in Math. 1140, Springer-Verlag, 1985.
[D2]
S. Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993), 39-60.
[D3]
S. Donkin, The q -Schur algebra, preprint 1996.
[El]
K. Erdmann, Symmetric groups and quasi-hereditary algebras, in: Finitedimensional algebras and related topics (Y. Dlab and L. L. Scott, eds.), Kluwer, 1994, 123-161.
[E2]
K. Erdmann, Tensor products and dimensions of simple modules for symmetric groups, Manuscripta Math. 88 (1995), 357-386.
[E3]
K. Erdmann, Decomposition numbers for symmetric groups and composition factors ofWeyl modules, J. Algebra 180 (1996), 316-320.
[G]
J. A. Green, Polynomial representations of GL n , Lecture Notes in Math. 830, Springer-Verlag, 1980.
[G2]
J. A. Green, Combinatorics and the Schur algebra, J. Pure Appl. Algebra 88 (1993),89-106.
[JK]
G. D. James, A. Kerber, The representation theory of the symmetric group, Enc. of Math. 16, Addison & Wesley, 1981.
84
Karin Erdmann
[11]
G. D. James, Representations of the symmetric groups over the field of characteristic 2, 1. Algebra 38 (1976), 280-308.
[12]
G. D. James, On the decomposition matrices of the symmetric groups I, J. Algebra 43 (1976),42-44.
[13]
G. D. James, The decomposition of tensors over fields of prime characteristic, Math. Z. 172 (1980),161-178.
[Ja]
1. C. Jantzen, Representations of Algebraic Groups, Academic Press, 1987.
[M]
S. Martin, Schur algebras and representation theory, Cambridge University Press, 1993. O. Mathieu, On the dimension of some modular irreducible representations of the symmetric group, Lett. Math. Phys. 38 (1996), 23-32.
[Ma] [P]
B. Parshall, Finite-dimensional algebras and algebraic groups, Contemp. Math. 82 (1989), 97-114.
[R]
C. M. Ringel, The category of modules with good filtrations over a quasihereditary algebra has almost split sequences, Math. Z. 208 (1990), 209-225. T. J. Rivlin, Chebyshev polynomials: From Approximation Theory to Algebra and Number Theory (2nd ed.), Wiley, 1990. 1. Schur, Ober die rationalen Darstellungen der allgemeinen linearen Gruppe (1927), in: 1. Schur, Gesammelte Abhandlungen III, 68-85, Springer-Verlag, Berlin, 1973. S. Xanthopolous, On a question of Verma about indecomposable representations of algebraic groups and their Lie algebras, PhD Thesis, London, 1992.
[Ri] [S]
[X]
Mathematical Institute 24-29 St. Giles Oxford OX1 3LB England Email:
[email protected] On Extended Block Induction and Brauer's Third Main Theorem G. Huang
1. Introduction As generalizations of Brauer correspondence in the study of the modular representation theory of finite groups, four different definitions of block induction have been proposed and used. They are Brauer induction [Fe], p-regular induction [B11], extended induction [Wh] and Alperin-Burry [AI]. The first three are defined in terms of central characters, while Alperin-Burry induction is defined in terms of module-theoretic properties of block ideals. Although the four definitions are different, they are closely related. Among them, extended block induction is the weakest. We examine whether properties of the other types of induction also hold for extended induction. For instance, we found a class of infinitely many examples to show that the transitivity which holds along blocks under Brauer induction and p-regular induction does not hold under extended induction. We also give some p-Iocal characterizations of extended block induction, discuss Brauer's Third Main Theorem in the extended induction sense, and establish an affirmative result on an aspect of Brauer's Third Main Theorem for p-solvable groups. That is, for p-solvable groups, principal blocks always induce in the extended sense to principal blocks if the induction is defined. Now let us fix our notation and state some definitions. Let G be a finite group. Let p be a prime number and let (F, R, K) be a p-modular system. Assume that F and K are splitting fields for every subgroup of G. Let (9 be R or F and let H be a subgroup of G. The Brauer map Br~ (or simply BrH): (9G -+ (9H is defined by Br~(x) = x if x E Hand 0 if x E G - H for all x E G and is linearly extended to (9G. If A is a G-algebra, let A H denote the set of H -fixed points. Let Gp (resp. G pi) denote the set of p- (resp. pi -) elements of G. For a set S ~ G, let F S denote the subspace spanned by S, and Z F S the intersection of F S with the center ZFG of FG. If b is an (9G-block, where "block" refers to block
86
G. Huang
idempotent, then Ab denotes the central character of FG associated with block b. If e = LrjlEIIT(G) arjl 3 be an even integer. Let G = GL(n, 2) and H =
[~
GL(n
~ 1,2)] ..::. GL(n, 2).
Let GF(2n - 1)# = (x). Then under the multiplicative action on the vector space GF(2 n - 1), x can be regarded as an element of H of order 2n - 1 - 1. Let q be a Zsigmondy prime number with respect to (2, n - 1). Namely, q is a divisor of 2 n - 1 - I but not a divisor of any 2' - 1 with r < n - 1. Let T = (z} E Sylq ((x). Then we have a chain of three groups:
T ni > ni+1 > m for 1 ::: i < t. In other words, in Step I we obtain a rank comparison between rank o (cab N p there exists c in 'B p with m = lel < Ibl such that there is an idempotent I of ( 3 and n i= 6, there is another central extension of Sn, usually denoted by Sn. However, the character theory of Sn may be described in exactly the same fashion as that of Sn [C, Section 3.9] and it is easy to see_that th~ results described in this section remain valid if we replace the group Sn by Sn.
References [A]
J. L. Alperin, Local representation theory, in: The Santa Cruz Conference on Finite Groups (B. Cooperstein and G. Mason, eds.), Proc. Symp. Pure Math. 37, Amer. Math. Soc., Providence, R.L, 1980, 369-375.
[B]
M. BroUl~, Theorie locale des blocs, in: Proceedings of the International Congress of Mathematicians, Berkeley, 1986, 360-368.
[C]
M. Cabanes, Local structure of the p-blocks of Sn, Math Z. 198 (1988), 519-543.
[H] [HH]
[J]
J. F. Humphreys, Blocks of projective representations ofthe symmetric groups, J. London Math. Soc. (2) 33 (1986), 441-452. P. N. Hoffman and J. F. Humphreys, Projective Representations of the Symmetric Groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1992. G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math. 682, Springer-Verlag, Berlin 1978.
100
Radha Kessar
[K]
Radha Kessar, Blocks and source algebras for the double covers of the symmetric and alternating groups, preprint, 1995.
[L]
Markus Lincklemann, The isomorphism problem for cyclic blocks and their source algebras, preprint, 1994.
[M 1]
A. O. Morris, The spin representation of the finite group, Proc. London Math. Soc. (3) 12 (1962), 55-76.
[M2]
A. O. Morris, On Q-functions,1. London Math. Soc. 37 (1962),445-455.
[0]
J. Olsson, On the p-blocks of the symmetric and alternating groups and their covering groups, 1. Algebra 128 (1989),188-213.
[P I]
L. Puig, Pointed groups and construction of characters, Math Z. 176 (1981), 358-369.
[P2]
L. Puig, Local fusions in block source algebras, 1. Algebra (1986),358-369.
[P3]
L. Puig, On Joanna Scopes' criterion of equivalence for blocks of symmetric groups, Algebra Colloq. I: 1 (1994),25-55.
[S]
I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitionen, J. Reine Angew. Math. 139 (1911), 155-250.
[Sc]
1. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), 441-455.
Department of Mathematics Yale University New Haven eT 06520 U.S.A.
A Survey on the Local Structure of Morita and Rickard Equivalences between Brauer Blocks Lluis Puig
1. Introduction 1.1. Since Jeremy Rickard's thesis - developing a "Morita theory" for the equivalences between the so-called derived categories of the categories of modules over two algebras [15], and exhibiting such an equivalence when the starting two algebras are the block algebras in characteristic p of Brauer blocks with the same cyclic defect gorups and the same inertial quotients - it has appeared an ample belief in the sense that the existence of such equivalences could "explain" the similarities between some pairs of Brauer blocks which, however, are far from being Morita equivalent. Of course, the first example is Rickard's equivalence for blocks with cyclic defect groups mentioned above, lifted by Markus Linckelmann to characteristic zero [6] and explicited by Raphael Rouquier which exhibits in [21] a quite simple two terms bicomplex inducing such an equivalence. Although, at present, there are not so many other examples, the reasonably expected derived equivalences exposed by Michel Broll(~ in [3], [4] and [5] justify, from our point of view, an effort to understand a priori the consequences of such equivalences between the socalled local structures [2], [1] and [8] of the concerned blocks, in particular seeking inductive constructions. A first attempt in that direction is Rickard's result on the so-called splendid equivalences in [18], which we improve here (Section 6). 1.2. But before trying to understand the relationship between the local structures of two blocks having equivalent derived categories of the categories of modules, it is prudent to start by analysing this relationship when they have already equivalent module categories; that is to say, in a widely employed terminology, when the blocks are Morita equivalent. Let us fix some notation; as usual, p is a prime number and eJ is a complete discrete valuation ring having an algebraically closed residue field k = (') / J (eJ) of characteristic p (we allow the possibility eJ = k), and all the modules we consider are finitely
102
L1uis Puig
generated over 1. In particular, they are locally finite. For some important locally finite varieties the problem of the existence of a finite base of laws remains open. For instance (see [1], question 4.48), the following problem is still open. Question 1. Is it true that any locally finite variety of groups with abelian Sylow subgroups is finitely based? Let us consider this problem. We will call such varieties A-varieties. First, two simple facts. Lemma 1. This problem has affirmative solution finitely based.
if each solvable A-variety is
So, we may consider only solvable A -varieties.
150
Sergei A. Syskin
Lemma 2. Let G be a finite group with abelian Sylow subgroups. Then the class of solvability of G is at most the number of prime divisors of IG I.
Let S be any solvable A -variety. Assume that S has no finite base of laws. Standard arguments show that S contains an infinite sequence of groups G I , G2, .. , with the following properties. (1) Every Gi, is a finite solvable group all of whose Sylow subgroups are abelian.
(2) All Gi have a restricted exponent and hence a restricted class of solvability by Lemma 2. (3)
Gi is not isomorphic to any factor of G) for i =j:. j (factor = factor-group of some subgroup).
(4)
G i is a critical group.
By definition, a group is critical if it does not lie in the variety generated by all its proper factors. Under these circumstances (Sylow sUbgroups in S are abelian) this means that every Gi has exactly one minimal normal subgroup Mi. Clearly, Mi is a p-group for some prime number p, and we may assume that p is the same for all Gi. Obviously, Pi = C (Mi) is a Sylow p -subgroup of G i. Hence G i/ Pi is a pi -group, that is of order prime to p. Moreover, we can assume that Pi = Mi' therefore
Thus, we have an ordinary faithful irreducible representation of Gi/ M i on
Mi. We have either to construct such a sequence G i satisfying (1)-(5), or to prove that it does not exist. If we are going to prove the nonexistence, we may assume something more. For example, we may assume that
(2 ') All Gi have the same exponent and the same class of solvability. Let Rand S be solvable finite groups of the same class n. We may define their derived subgroups: R(l) = R, R(i+I) = [R(i), R(i)]. We say that R and S are of the same type if the exponent of R(i) / R(i+I) equals the exponent of S(i) / S(i+I) for all i = I, ... ,n. Clearly, we can assume that (6) All Gi are of the same type.
Locally Finite Varieties
151
Using induction on the type, we can assume that our conjecture holds for the sequence G I Mi. It is a simple matter to prove that the last sequence has an infinite subsequence which is a chain of groups. Hence we may assume that (7)
Gi + I I Mi + I has a subgroup isomorphic to G i/ Mi for all i = 1, 2, ....
Of course, one may assume something else. For example, in the definition of type one can consider the series R(i) instead of RU) where R(i)1 RU+I) is the (unique) largest abelian normal subgroup of RI RU+I). Perhaps, this alternative definition is more helpful since all ordinary representations under consideration are monomial over some extension of G F(p). Thus, we may propose the following
Question 2. Let RI .::: R2 .::: '" be an infinite chain of finite solvable groups with abelian Sylow subgroups. Assume that all Ri have the same class of solvability and the same exponent. Let p be a prime which does not divide this exponent. Let Mi be a fixed faithful irreducible G F (p) Ri -module. Is it true that one of these groups, say Rj, contains a subgroup H isomorphic to Ri, i i= j, such that, for some H -submodule V of Mj, two natural semidirect products Ri Mi and V H are isomorphic? Question 3. Let p be a prime, A p the variety of all abelian groups of exponent p, and let N be a locally finite nilpotent variety such that p does not divide the exponent of N. Is it true that any subvariety of the variety ApN is finitely
based? It is known that any nilpotent variety has a finite base of laws, see [2].
References [1]
Kourovka notebook, A collection of unsolved questions in group theory, 11th edition, Novosibirsk 1990.
[2]
H. Neumann, Varieties of groups, Springer-Verlag, 1967.
[3]
A. Yu. Ol'shanskii, On the problem of a finite basis of identities in groups, Math. USSR-Izvestija 4 (1970),381-389.
Reinsurance Group of America 660 Mason Ridge Center Drive 5t Louis MO 63141
U.5.A.
Contains articles bymanyof theparticipants at theConference on Modular Representation Theory which tookplaceunder theauspices of theMathematical Research lnstitute of theOhioStateUniversity withadditional support fromthe National Foundation.The Science conference celebrated thecareer of Professor V/alter Feitof Yale University ontheoccasion of his65thbirthday. lt alsoroughly coincided withthe centenary of thetheoryof groupcharacters. Theparticipants included mostof the leaders in thefieldof finitegrouprepresentations andthisvolumecontains and expands on manyof thestimulating lectures fromtheconference. Thisvolume is addressed to specialists andstudents in thefieldof grouprepresentation theory, including boththegeneral theory andthespecial theories related to the groupsandthefinitegroupsof Lietype.Latest representations of thesymmetric reportsaregivenon important conjectures including theLusztig conjecture, the Alperin-McKay-Dade conjectures andthekGV)problem. lmportant newresearch avenues areilluminated, including thetheory of infinite-dimensional modules forfinite groupsandthetheoryof Rickard equivalences of modulecategories, Bothexperts andgraduate students willfindthebooka fascinating introduction to a widearrayof problems newtechniques exciting and attheresearch level.
rssN0942-0363 rsBN3-11-015806-X
