Representation Theory of Finite Groups: Proceedings of a Special Research Quarter at the Ohio State University, Spring 1995 9783110806298, 9783110158069

Table of contents :
Preface
Table of Contents
Embeddings, Geometries and Representations: Connections and Computations
Infinite Dimensional Modules for a Finite Group
Degrees and Diagrams of Integral Table Algebras
Canonical Induction Formulae and the Defect of a Character
Counting Characters in Blocks, 2.9
The Defect Groups of a Clique
Representations of GLn(K) and Symmetric Groups
On Extended Block Induction and Brauer's Third Main Theorem
On Blocks and Source Algebras for the Double Covers of the Symmetric Groups
A Survey on the Local Structure of Morita and Rickard Equivalences between Brauer Blocks
Some Open Conjectures on Representation Theory
Are All Groups Finite?
Locally Finite Varieties of Groups and Representations of Finite Groups

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Ohio State University Mathematical Research Institute Publications 6 Editors: Gregory R. Baker, Walter D. Neumann, Karl Rubin

Ohio State University Mathematical Research Institute Publications 1 2 3 4 5

Topology '90, B. Apanasov, W. D. Neumann, A. W. Reid, L. Siebenmann (Eds.) The Arithmetic of Function Fields, D. Goss, D. R. Hayes, Μ. I. Rosen (Eds.) Geometric Group Theory, R. Charney, M. Davis, M. Shapiro (Eds.) Groups, Difference Sets, and the Monster, K. T. Arasu, J. F. Dillon, K. Harada, S. Sehgal, R. Solomon (Eds.) Convergence in Ergodic Theory and Probability, V. Bergelson, P. March, J. Rosenblatt (Eds.)

Representation Theory of Finite Groups Proceedings of a Special Research Quarter at The Ohio State University, Spring 1995

Editor Ronald Solomon

w DE

Walter de Gruyter · Berlin · New York 1997

Editor RONALD SOLOMON

Department of Mathematics, The Ohio State University 231 West 18th Avenue, Columbus, OH 43210, USA Series Editors: Gregory R. Baker Department of Mathematics, The Ohio State University, Columbus, Ohio 43210-1174, US A Karl Rubin Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA Walter D. Neumann Department of Mathematics, The University of Melbourne, Parkville, VIC 3052, Australia 1991 Mathematics Subject Classification: Primary: 20C Secondary: 20C15, 20C20, 20C30, 20C33 Keywords: 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 3-11-015806-X (alk. paper) 1. Finite groups - Congresses. 2. Representations of groups - Congresses. I. Solomon, R. C. (Ronald C.) II. Series. QA177.R46 1997 512'.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 3-11-015806-X

© Copyright 1997 by Walter de Gruyter & Co., D-10785 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 E X files: I. Zimmermann, Freiburg. Printing: Werner Hildebrand, Berlin. Binding: Lüderitz & Bauer G m b H , Berlin. Cover design: Thomas Bonnie, Hamburg.

To Walter Feit on the occasion of his sixty-fifth birthday

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 William 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 Walter 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 Walter Feit of Yale University on the occasion of his sixty-fifth birthday. It was a pleasure to bring together many of Walter'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:

viii

Preface

Jonathan L. Alperin, University of Chicago, On endo-permutation modules·, Matthew Κ. Bardoe, Imperial College, University of London, Representations, embeddings and geometries; David G. Benson, University of Georgia, Cohomology of modules for a finite group; Robert Boltje, University of Augsburg, Canonical induction formulae and the defect of a character; Michel Broue, 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 Β rauer 's First Main Theorem', and

Karin Erdmann, University of Oxford, Representations of symmetric groups GLn(K); Walter 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 algebras for 2Sn and 2 An; 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 of finite groups', J. 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

Μ. Κ. Bardoe Embeddings, Geometries and Representations: Connections and Computations

1

D. J. Benson Infinite Dimensional Modules for a Finite Group

11

Η. 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 GLn{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 Μ. 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 ρ - l o c a l structure of Lie-type groups over fields of characteristic ρ 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, Co\, and F124. Recently, through the w o r k of Quillen, B r o w n , and A d e m and M i l g r a m , e.g. [ A M ] and others, it has b e c o m e clear that understanding the geometries associated to a group m a y also be helpful in computing and understanding the c o h o m o l o g y of that group at a specific prime. Representation theory is also able to reap the benefits of this viewpoint. W h a t w e describe here is a t w o - w a y street between representation theory and geometry. O n e w a y is the method of constructing from a representation of a group a geometry for that group. This direction is w e l l known and easily understood. T h e other direction is more complicated. It involves inductively creating modules, based on a geometry associated to a particular group, f r o m modules f o r its subgroups. W e g i v e a short outline of what follows. First, w e define geometry and give examples of geometries for some familiar groups. Then w e outline some of the theory which describes how to construct representations w h i c h are described by geometries. W e attempt to explain w h y this process is interesting to both geometers and to representation theorists. Finally w e give a partial list computations and constructions w h i c h have been completed.

2

Μ. Κ. Bardoe

Geometries We take the following to be as our definition of geometry: Definition 1: Geometry. A geometry of rank η is an ordered sequence Γ = (Γο, Γ ι , . . . , Γ„_ι, *) of η pairwise disjoint non-empty sets Γ,· together with a symmetric incidence relation, *, on their union such that if F is any maximal set of pairwise incident elements then | F Π Γ; | = 1 for each i. These first two examples demonstrate how representations have been used to construct geometries. Example 1: The Projective Geometry for an «-dimensional vector space V, denoted PG(V), is the η — 1 sets given by the 1-spaces, 2-spaces,..., η — 1 -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 1-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, [RSI]. This geometry is based on the Steiner system 5(5,8,24). In the terminology of our definition Γ = (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 Γ,. Then define η-simplices to be «-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.

Figure 1: Geometry for SL3(2) viewed as simplicial 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-local geometries for the sporadic groups. Simplistically, we construct a geometry by taking the conjugacy classes of maximal subgroups containing a Sylow ρ-subgroup to be the objects of our geometry.

4

Μ. Κ. Bardoe

Then say that two such subgroups are incident if their intersection contains a common Sylow ρ-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 ρ-local 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 7 on Γ assigns to each simplex σ e Γ a representation 1a of Stab(a) c Aut(r). If τ is a face of σ there is a linear connecting map φ(jx : 3σ -»· J T such that φρσ ο φστ = φρτ whenever this composite map is defined. The 2fa and φστ are required to be Aut(r)-equivariant in the following sense: For each g e Aut(T) there is a mapping g : 3 1 such that gh = gh, %g = ( T ) g , and g ο φσ8,τ8 = φστ ο g.

The homology groups of such a sheaf 7 are k Aut(r)-modules. In particular, if we assume that the terms of 7 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 3 through the submodule structure of V = //o(3~)· Namely, if σ, r are simplices of the simplicial complex derived from Γ, then there exist submodules V„ of V upon restriction Stab(a) suchthat Va = . Also, if r C σ, then Va C Vz. Note that the star of a simplex, St(a), is also a simplicial complex, and that a sheaf, on Γ defines a sheaf, S, for St(a). Therefore if one understands the zero homology of sheaves for St(a) for σ of dimension < rank(T) — 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, Mnj, with connecting maps, φα,πι, then one may construct a universal sheaf, U, such that the module at any chamber is isomorphic to Mc and U7Ii = Mnj, and the modules at other simplices are defined to be tfo(St(a)).

Embeddings, Geometries and Representations

5

Example 4: [RS2] Let Γ be PG(V) where V is a 3 dimensional space over F2. Then Aut(T) = SLI(2). Suppose we define a chamber generated sheaf U by assigning to a chamber a fixed 1 -space. The face of a chamber associated to a point is assigned a fixed 1 -space, and the face of a chamber associated to a line is assigned a 2-space which contains the three fixed 1-spaces of the chambers which contain the line. In the terminology of our definition of a sheaf, Μ ρ 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 line stabilizer, and the connecting maps are inclusion maps. Then Ronan & Smith show that Ho (It) = V for the sheaf defined by these conditions. Another result of Ronan & Smith, [RS2], shows that if one can form a sheaf, J , from the submodule structure of a module, V, then V is a quotient of HQCJ). Therefore from the last example we see that V is the only module satisfying the condition described by that sheaf, as ffo(U) is an irreducible module for 5L3(2). Therefore this result about the form Ηο(7) provides a kind of local recognition result for SLT,(2) modules. And in general computation of HO(T) 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 Γ is an injective incidence preserving map, it, from Γ to PG{V), the projective geometry of a vector space over the appropriate field, such that the points are mapped into the 1 -spaces of V and the lines are mapped into the 2-spaces of V.

6

Μ. Κ. 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. 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 Example

Γ = (special 1-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 Γ, we rephrase this question into the language of sheaf theory in the following way: What modules support the sheaf, given by assigning a 1 -space to the point stabilizer and a 2-space to the line stabilizer? From what we have said above one such module is Hq{7). 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 1 -space. Therefore we can define an embedding as a map π' from Γ 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 +

vr

=

0 for

I — { 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 vp + vq + vr — 0 for / = {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, DOT Ε ΑΊ

Near-hexagon

M24

Near-hexagon

3)

U4(

Near-octagon

h. 2

Near-hexagon

3

Near-hexagon

U6( 2).2

Near-hexagon

07(2)

Involution geometry

UA(3)

Involution geometry

Suz

Involution geometry

Co ι

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 ι He Ru M2 4 3.^4(2) He

^4(2)

Μ

Ss Aut(M 22 ) 3. Aut(M 22 ) M2 3 C02 3 -".On H

BM 3

4I/I

.BM

Univ. Embedding Natural Module Adjoint Module 0 11 i©TT 20 1 26 1Θ1 26 1Θ1 20 1®1 8 τ τ 34 34' 1 ® 1 142 1 274 1 ® 1 ®24 24 51 28 11 6φ 5 52 0 6 11 12® 11 0 23 23 0 0 0

Remarks [RS2] [SV], [V] [RS 3] [RS3] [Y] [FS] [FS] [Y] [Y] [B3] [B2] [Bl] [Sm] [MS] [MS] [IS2] [IS2] [IS2] [IS2] [IS1] [IS1] [IS1] [IS1] [IS3] [Sh] [IS2] [IS2]

8

Μ. Κ. Bardoe

Bibliography [AM]

A. Adem and R. J. Milgram, The cohomology of the Mathieu group M22, Topology 34 (1995), 389-410.

[Bl]

Μ. K. Bardoe, The universal embedding for the Co\ involution geometry, in preparation, 1995.

[B2]

Μ. K. Bardoe, The universal embedding for the Suzuki sporadic simple group, Preprint, accepted to J. Algebra, 1995.

[B3]

Μ. K. Bardoe, The universal embedding for the i/4(3) involution geometry, Preprint, accepted to J. Algebra, 1995.

[FS]

D. Frohardt and S. Smith, Universal embeddings for the 3 0 4 ( 2 ) hexagon and the J2 near-octagon, Europ. J. Combin. 13 (1992), 455-472.

[IS 1 ]

A. A. Ivanov and S. V. Shpectorov, Geometries for sporadic groups related to the Petersen graph. II, Europ. J. Combin. 10 (1989), 347-361.

[152]

A. A. Ivanov and S. V. Shpectorov, The flag-transitive tilde and Petersen-type geometries are all Known, Bull. Amer. Math. Soc. 31 (1994),173-184.

[153]

A. A. Ivanov and S. V. Shpectorov, Natural representations of the /'-geometries of Co 2 -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, N.J. 1974.

[MS]

G. Mason and S. Smith, Minimal 2-local geometries for the Held and Rudvalis sporadic Groups, J. Algebra 79 (1982), 286-306.

[RSI]

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 Chevalley Groups, J. Algebra 96 (1985), 319-346.

[RS3]

M. A. Ronan and S. D. Smith, Computation of 2-modular sheaves and representations for £4(2), Λ7, 3S 6 , and Μ24, 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.

[Sm]

Stephen D. Smith, Universality of the 24-dimensional embedding of Comm. Algebra 22 (1995), 5159-5166.

[SV]

S. Smith and H. Völklein, A geometric presentation for the adjoint module of SLi(k), J. Algebra 127 (1989), 127-138.

[V]

H. Völklein, On the geometry of the adjoint representation of a Chevalley group, J. Algebra 127 (1989), 139-154.

Co\,

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 2B7 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 Μ for Z2 χ Z2 in characteristic two, which satisfies Μ = Μ φ Μ φ Μ but not Μ = Μ φ Μ. 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 ρ-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. Then for some λ in k(t), the linear map 1 / - λ Identity : k(t) * V

k(t) ®k 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 χ act as the linear transformation / . If / — λ. Identity is an isomorphism for all λ 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 ® χ — t ® 1 is not an isomorphism on k(t) £ 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 / and g give isomorphisms from k(t) V to k(t) k W, then the vector spaces V and W are 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 Ε = ( g i , . . . , gr) be an elementary abelian group of order pr. Let X, be the element g, — 1 of the group algebra kE, so that X? = 0 and X,· is in the Jacobson radical J{kE). Let Vg(k) denote the quotient space J(kE)/J2(kE), and let x\,..., xr be r the images of X\,..., Xr in V E(k). It is not hard to show that they form a basis for this quotient space. Let yi,... ,yr be the linear functions Vg(k) -*• k given by yt(xj) = 1 if i = j and zero otherwise. Then regarding VrE(k) as an affine space, its coordinate ring is the polynomial ring &[;yi,..., If a = λι*ι Η

l-krxr € VrE{k)

Infinite Dimensional Modules for a Finite Group

13

is not equal to zero, then we set ua = 1 +λιΧι

+ ··· +

krXr,

an element of order ρ in the group algebra kE. Theorem 2.1 (Dade's Lemma). Suppose that k is algebraically closed. If Μ is a finitely generated kE-module such that the restriction of Μ to (ua) is free for each point 0 φ a e VE(k), then Μ is a free k Ε -module. The hypothesis that Μ is finitely generated is certainly necessary here. There are examples of infinitely generated modules for Ζ/2 χ Z/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 [Be3]: Theorem 2.2. Let Κ be an algebraically closed extension of k of transcendence degreee at least r — 1, and set VE(K) = J{KE)/J2(KE). If Μ is a kE-module such that (Κ Μ) φ(Μα> is free for all nonzero a € Vg(K), then Μ is a free kE-module. The idea of the proof is to reduce to the rank two case, and then choose two linearly independent elements η, η' e Hl(E,¥p)

=

J(¥pE)/J2(FpE),

so that the Bocksteins β(η) and β (η') are algebraically independent elements of H2(E,¥P). They induce maps f,g:E$E(k,

M)^Ert2kE(k,

M)

with the property that for all λ, μ 6 Κ, not both zero, kf + ng: Εχ?ΚΕ(Κ, K®kM)

&KE(K,

Κ ®k Μ)

is an isomorphism. It now follows from the vector space lemma discussed earlier, that Ext° kE (k, M) = 0, which implies that Μ is free.

14

D. J. Benson

3. The Rank Variety For Μ a finitely generated kE-module, with k algebraically closed, Carlson's definition [Ca] of the rank variety of Μ is VrE(M) = {0 φ a € VrE(k) I Μ | { Μ α ) is not free} U {0}. This is a closed homogeneous subvariety of VE(k), and Dade's lemma may be interpreted as saying that Μ is free if and only if VrE{M) = {0}. In fact, the dimension of VE ( Μ ) determines the polynomial rate of growth of the minimal free resolution of Μ as a k Ε -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 Μ is periodic (after the first term, which may be too big because of the free summands of Μ ) . 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 d o i s j u s t t o l o o k a t VE(K kM), where Κ 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 Κ is an extension of k of transcendence degree at least r . Recall that if V c VE (k) is a closed irreducible subvariety, then the set Ρ = {/ € k[y\,..., is a prime ideal in

yr] \ f vanishes on V}

, . . . , yr], and k[V] =

k[yu...,yr]/P

is the coordinate ring of V. It is an integral domain, and its field of fractions k(V) is the function field of V. It is generated as an extension field of k by the images y\,... ,yr of the elements j i , . . . , yr. The transcendence degree of k(V) over k is equal to the Krull dimension of and is by definition the dimension of the variety V. Since Κ is an algebraically closed extension field of k of transcendence degree at least as big as that of k(V), it follows that the inclusion of k into Κ extends (not by any means uniquely) to an embedding of k(V) into K. Let

Infinite Dimensional Modules for a Finite Group

15

t\,... ,tr be the images of j i , . . . , yr under such an embedding. The generic point of V is defined to be the element + · · · + yrxr €

vrE(k(V)).

Note that this point is well defined, independently of the chosen basis for VE(k). A point which is of the form t\x\+---

+ trxr e VrE(K)

for some embedding of k(V) into Κ as above, is said to be a generic point of V over K. If t\x\ + b trxr is any point in VE(K), set p c k[y\,..., )v] equal to the ideal consisting of all polynomial relations satisfied by t\,..., tr over k. Since AT is a field, this ideal is necessarily prime. Let V be the associated subvariety of VrE{k). Then the point t\x\ + ... + trxr is a generic point of V over K. So every point defined over Κ 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 t\x\ Η 1-t r x r is generic for some inhomogeneous subvariety V, then the dimension of V is less than r, so there is an element λ e Κ which is algebraically independent of t\,... ,tr. Then the point kt\x\ Λ b Xtrxr is generic for the homogeneous hull of V, namely the smallest homogeneous subvariety containing V. Now if Μ is a k Ε-module (not necessarily finitely generated), then the question of whether (Κ Μ) 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 VrE(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 ) 1. Let M2 be the kΕ-module with generators m\,m'2,... and relations X% m^ = X\m'j+l for each / > 1. The modules M\ and M2 are superficially similar, but M\ has complexity one, while M2 has complexity two. The set VE{M\) consists of just a single line through the origin, while YE(M2)

= VrE(k)\VrE(M

1).

Let φ : k —> M2 be the homomorphism taking the basis element of k to the element Xim[. Then the cokernel of φ is isomorphic to M\. We can modify this example by replacing the actions of X\ and X2 by aX 1 + bX2 and cX\ + dX2,

where

has the effect of moving the distinguished line through the origin by the corresponding linear transformation on VE(k). If the resulting line corresponds to the point (λ : μ) e Ρ1 (k), let us write Μ(χ:μ) for the module obtained from M2 in this way. Up to isomorphism, it only depends on the point (λ : μ) € Ρ 1 (A:). Let φ(χ:μ) be the corresponding homomorphism from k to Μ(λ:μ). Then we may put these together to form a map k

Μ Θ (λ:μ)· 1 (λ:μ)εΡ (jfc>

Infinite Dimensional Modules for a Finite Group

17

Let Μ be the cokemel of this map. Then Μ is projective on restriction to r (.ua) for every nonzero € but Μ itself is not projective. In some E sense, what we have done is to take affine 2-space, and remove all the lines through the origin. However, what we haven't done is to remove lines through the origin defined over a larger field. So the generic point of the affine plane is still in the variety. In fact, we have

a V (k),

ye{M) = {vrEm. References [Bel ]

D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196-205.

[Be2]

D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, Math. Proc. Cambridge Philos. Soc. 118 (1995), 223-243.

[Be3]

D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules II, Math. Proc. Cambridge Philos. Soc. 120 (1996), 597-615.

[Be4]

D. J. Benson, J. F. Carlson and G. R. Robinson, On the vanishing of group cohomology, J. Algebra 131 (1990), 40-73.

[Ca]

J. F. Carlson, The varieties and cohomology ring of a module, J. Algebra 85 (1983), 104-143.

[Da]

E. C. Dade, Endo-permutation modules over ρ-groups II, Annals of Math. 108 (1978), 317-346.

Department of Mathematics University of Georgia Athens, G A 30602 USA Email: [email protected]

Degrees and Diagrams of Integral Table Algebras Harvey I. Blau

Integral table algebras satisfy fundamental properties abstracted from group algebras and character rings of finite groups, and from the adjacency algebras of commutative association schemes. Implicit in the work of Schur [Sc], these structures, or variations of them, have been defined and studied independently under several guises, including "C-algebras" [K, BI], "pseudo groups" [Br], and "hypergroups" (as in [McM]), as well as "table algebras" [Al, AF, Bll]. A connection has been observed recently [B12] between table algebras and certain diagrams, among which the affine diagrams of Lie theory and combinatorial geometry are special cases. We present here an exposition of this link, give some examples, and state some results which show how the structure of an integral table algebra, and of its associated diagram, can be determined in certain situations by information on the "degrees" (defined below) of its distinguished basis elements. The most recent results, Theorems 2 and 3 below, are joint work of Bangteng Xu and the author. Throughout, C denotes the complex numbers, R the reals, and R + the positive reals.

Definition. [Al, AF, A2, Bll] Let Β = {b\, 02,..., bk} be a basis of a finite dimensional, associative and commutative algebra A over C, with identity

element 1 ^ = b\. Then (Α, B) is a table algebra (and Β is a table basis) if and only if the following hold:

(I)

For all i, j, m, bibj = Σ * =1 βijmbm, with ßijm € M+U {0}.

(II)

There is an algebra automorphism (denoted by " ) of A whose order divides 2, such that bi e Β implies bt e B. (Then i is defined by

bj = bi, and bi e Β is called real if i = i.) (III) For all i, j, ßij\ φ 0 if and only if j = i.

20

Harvey I. Blau

If (Λ, Β) is a table algebra, then there exists a unique algebra homomorphism / : Λ C such that /(&,·) € M+ for all bt e Β [Al, Lemma 2.9; Bll, Proposition 2.11]. We fix the notation / for this homomorphism, and call the values /(£>,•) the degrees of (A,B). A table algebra (A, B) is called integral iff all structure constants ßijm and all degrees /(&;) are rational integers. Any finite group G yields two examples of integral table algebras: (Z(CG), Cla(G)), the center of the group algebra, with table basis the set of sums C of G-conjugacy classes C, with automorphism ~ extended linearly from inversion in G, and with degrees / ( C ) = |C| for all C e Cla(G); and (Ch(G), Irr(G)), the ring of complex valued class functions on G, with table basis the set of irreducible characters of G, with automorphism ~ extended linearly from complex conjugation of characters, and with degrees / ( χ ) = χ(1) for all χ € Irr(G). Another example is the adjacency algebra, or Bose-Mesner algebra, of a commutative association scheme [BI, Section II.2], Here, the table basis consists of the adjacency matrices corresponding to the defining relations of the scheme, the automorphism ~ is matrix transpose, and the degree of each adjacency matrix is its valency (row sum). There is an explicit theory of substructures, quotient structures and homomorphisms for table algebras [Bll], see also [BI, Section II.5; AF, Section 2]. The theory is important in the proofs of Theorems 2 and 3 stated below, and of related results, but there is no need to discuss it here. We note only that two table algebras (A, B) and (U,V) (or Β and V for short) are called exactly isomorphic iff there is an algebra isomorphism between A and U which restricts to a bijection between Β and V. In other words, Β and V, under suitable orderings, yield the same structure constants. A number of concepts from finite groups have generalizations to an arbitrary table algebra (A, Β). An element b e Β is called faithful iff t Supp B (fc") = Β [Al]. If G is a finite group and b = C € Cla(G), then b is faithful iff (C) = G. If b = χ € Irr(G), then b is faithful iff χ is a faithful character in the usual sense. An element b e Β is termed linear iff Suppß(£>") = {1} for some η > 0 (iff Supp B (bb) = {1}) [Al]. If b = C e Cla(G) then b is linear iff C c Z(G) iff |C| = 1. If b = χ € Irr(G), then b is linear iff χ(1) = 1. There are several useful ways to construct new table algebras from old. One of them is called rescaling [Al, Bll]: given table algebra (A, B), choose λ ι , λ2, . . . , Xk e Κ + , subject only to λι = 1 and λ· = λ,· for all i. Let B' = {kibi I bi e B}. It is easily seen that (A, Β') is another table algebra, with respect to the same automorphism " and homomorphism / .

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 £ B, define bf := bi if bt is real, tf = b? := bt + bj if not. Let B° := {bf I bi e B} and A~ : = {a e A | a = ä}. Then ( A ~ , B ° ) , the symmetrization of ( A , Β), is a table algebra with all elements real, and is

integral if (A, Β) is. Let (A, Β) be a table algebra with Then for 1 < i < k,

Β

=

{b\

=

1, b 2 , . . . , bk}. Fix

b

e

B.

k bbi

=

Y ^ d j i b j , j=ι

for unique dß e M + U {0}. Definition. [B12] (See also [BI, p. 114].) The representation graph of Β with respect to b (denoted Γ b(B)) is the directed graph with vertex set {1,2 (in bijection with Β), and where there is a directed edge from vertex i to j (which is labeled by d j i ) iff dji > 0. It is clear that Γb(B) is connected if and only if b is faithful. If b is real, then dij > 0 dji > 0, and so Γb(B) may be presented as an undirected, labeled graph, where each pair of adjacent vertices and corresponding edges appears as {dij,

i

djj) j

and whenever da > 0, there is a loop: »

(da).

Suppose that (A, Β) is an integral table algebra. If there exists b e Β which is faithful, real and has degree f(b) = 2, then Γb(B) is the underlying graph of a generalized Cartan matrix (as in [HPR]), on whose index set { 1 , 2 , . . . , / : } there is an additive function given by the values of / on Β [B12, 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 [B12, Theorem 2]. Exactly 13 of the 20 generalized Euclidean diagrams occur as representation graphs of integral table algebras with a faithful real basis element of degree 2. The 1 diagrams which do not occur are (in the notation of [HPR]) An, A\2, Bn, Ln, BLn, CDn and G 21. 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 [B12, 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 2 m , for any m > 0 [B12, Theorem 1]. 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(2η -(-1), be the subset of Cla(G) which consists of those class sums C (jc) = Z2n+i, the unique subgroup of index 2. Let A = (B), Then Γb(B) is CL n , where all nonzero dji — 1 except as degrees are as listed:

η > 0. Let Β such that C c b = χ + x~{. noted, and all

1 (2,1) b 1

2

2

2

2

2

Example 2. Let (A,B) = (Ch(G), Irr(G)) for G = SL(2,5). If b is an irreducible character of degree 2, then Γ^(β) is Eg, with all nonzero dji = 1 and all degrees as listed: 3

1 1

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 of the 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. [BX1] A table algebra (A, Β) is called homogeneous (of degree λ) iff \B\ > 1 and, for some fixed λ e R+, f(b) = λ for all b e B\{1}. Example 1, 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 λ. This means that a complete classification of homogeneous integral table algebras is rather a tall order. The degree λ which results from rescaling an arbitrary integral table algebra, as in Theorem 2, is usually very large. Example 3. [BX1] Fix λ € Ε with λ > 1. Let a = (λ - l)/2, β = (λ + l)/2. For each integer m > 0, we define an integral table algebra (Α,Γ„(λ). Let Tm(X) := {1 , xo, χι,..., Xm} be a basis for an (m + 2)dimensional vector space A over C, and define products of these vectors so that 1 is the multiplicative identity, and with, for 0 < i, j < m, XiXj

—

axi+j + ßxi+j+i λ · 1 + axo + axm axi+j-m + ßxi+j-m-1

if i + j < m, if i + j = m, if i + j > m.

Then (A, Tm(X)) is a table algebra, with i ; = xm~i for all i, and which is homogeneous of degree λ. If λ > 3 is an odd integer, then (A, Tm(k)) is integral. If m is even, then xmß is real and faithful, and the representation graph rXm/2(Tm(k)) is

24

Harvey I. Blau

We consider briefly the symmetrization (A , Τ η ( λ ) ° ) . Order the basis as {1, Xm/2,X0 +Xm,Xm/2-1

+

2+1 >*1 + *m-l, Xm/2-2 + *m/2+2, · · ·} ·

The multiplication of these elements by xmß matrix of structure constants 0 λ 0 0 0 0

1 0 2a 0 0 0

0 a 0 ß 0 0

0 0 ß 0 a 0

0 0 0 a 0 ß

0 0 0 0 β 0

0 0 0 0 0 a

0 0 0 0 0 0

a

0

ß

=

yields the tridiagonal

where δ = a or β, as m = 0 or 2 (mod 4). Then the graph Γχο

(λ, I) 1

(2α, a) (0,0) (a, a) * 2λ 2λ 2λ

(3. J) 2λ

(a, a) 2λ

2λ

({,/) 2λ

(T m (X)°)

(λ-ί,Α-ί) 2λ

Thus, ( A - , r m ( A ) ° ) , for m even, is a table algebra of Ρ-polynomial type [BI, p. 317], that is, each element of Tm(k)° is a polynomial in xm/2-

Degrees and Diagrams of Integral Table Algebras

25

The Tm(λ)° do not arise from association schemes, as in [BI, Section III.l], since the sequences of subdiagonal entries and of superdiagonal entries are not monotonic (see [BI, Proposition III. 1.2]). The existence of these Ρ-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 Ρ-polynomial type seems highly nontrivial. Example 4. Let Η be an abelian group which admits a fixed-point-free action by Ζn (cyclic of order η), for some η > 0. Let G be the semi-direct product HZn, let Β consist of all C e Cla(G) such that C c H, and set A = {B). Then (A, Β) is an integral table algebra which is homogeneous of degree n. (Example 1 is clearly a special case of Example 4.) Remark. The table algebras Tm(k) 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~\ = a x m - i +β*ο holds in Tm(X), whereas there existnoelements b, c in Cla(G) or Irr(G) with c φ b,b φ b, and SuppB(fcc) = {c,b} [Al, Corollary Ε ' ] . We recently have determined all integral table algebras (A, B) which are homogeneous of degree 3, and for which Β contains a faithful, real element [BX2]. The full result is too detailed to give here, so we restrict to the case where Β contains no nontrivial linear elements. First, we need to introduce a few more table algebras. Example 5. Three table algebras, with table bases V2, V3, V4 resp., are defined by the following products of nontrivial basis elements. V2 :— {1, υ}, where v2 = 6-1 + v; V3 := {1, υι, V2), where vf = 3-1 + 2v2i V1V2 = V2V\ = 2vi + V2, vl = 3- \ + Vi + V2\ V4 := {1, i>i, V2, v^}, where v2 = 3-1 + v\ + v2 =

v\ = 3-1 + 2v2,

V\ V2 = V2V1 = Vl + 2V3, V1V3 -- V3V1 — 2v2 + V3, V2V3 = V3V2 = 2v\ + υ3· 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

T h e o r e m 3 [ B X 2 ] . Let (A, Β) be geneous of degree 3, and such that nontrivial linear elements. Then Β or Tm (3) for some even integer m

an integral table algebra which is homoΒ contains a faithful real element and no is exactly isomorphic to one of V2, V3, V4 > 0.

A r a d , F i s m a n , M u z y c h u k and Miloslavsky, in recent w o r k [ A F M ] , have obtained s o m e results on h o m o g e n e o u s integral table algebras of degree 3 w h i c h have n o nontrivial real e l e m e n t s in the table basis. M a n y instances of E x a m p l e 4 are of this type, as is Tm(3) f o r m odd. T h e m a i n t h e o r e m of [ A F M ] characterizes the algebras of E x a m p l e 4 w h e n η = 3 and is odd.

References [ΑΙ]

Ζ. Arad and Η. I. Blau, On table algebras and applications to finite group theory, J. Algebra 138 (1991), 137-185.

[A2]

Z. Arad, Η. I. Blau, J. Erez and E. Fisman, Real table algebras and applications to finite groups of extended Camina-Frobenius type, J. Algebra 168 (1994), 615-647.

[AF]

Z. Arad and E. Fisman, On table algebras, C-algebras and applications to finite group theory, Comm. Algebra 19 (1991), 2955-3009.

[AFM] Z. Arad, E. Fisman, V. Miloslavsky and M. Muzychuk, On antisymmetric homogeneous integral table algebras of degree 3, Bar-Ilan University preprint. [BI]

E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/ Cummings, Menlo Park, 1984.

[Bll]

Η. I. Blau, Quotient structures in C-algebras, J. Algebra 175 (1995), 24-64.

[B12]

Η. I. Blau, Integral table algebras, affine diagrams and the analysis of degree two, J. Algebra 178 (1995), 872-918.

[BX1] Η. I. Blau and B. Xu, On homogeneous integral algebras, J. Algebra, to appear. [BX2] Η. I. Blau and Β. Xu, Homogeneous integral table algebras of degree 3, Northern Illinois University preprint. [Br]

R. Brauer, On pseudo groups, J. Math. Soc. Japan 20 (1968), 13-22.

[HPR] D. Happel, U. Preiser and C. M. Ringel, Binary polyhedral groups and Euclidean diagrams, Manuscripta Math. 31 (1980), 317-329. [Κ]

Y. Kawada, Über den Dualitätssatz der Charaktere nichtcommutativer Gruppen, Proc. Phys. Math. Soc. Japan (3) 24 (1942), 97-109.

[McK] J. 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.I., 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

[Sc] [SI]

27

I. Schur, Zur Theorie der einfach transitiven Permutations-gruppen, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl, 1933, 598-623. 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.S.A. Email: [email protected]

Canonical Induction Formulae and the Defect of a Character R.

Boltje

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 /-defect ά(χ) of an irreducible character χ of a finite group G by mapping χ precisely to the d(x)-th layer in the filtration A c l~l A c l~2A ..., 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. [Sn88], and the author, cf. [Bo89], 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-°(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±°(G) c R(G) is the set of all non-negative integral linear combinations of elements in Irr(G). Furthermore, let R+(G) be the free abelian group on G-conjugacy classes [Η, ψ]ο of the poset M ( G ) of pairs (Η, φ) where Η is a subgroup of G and ψ is a one-dimensional character of Η . The poset structure on M ( G ) is defined by (Κ, ψ) < (Η, φ), if and only if Κ < Η and ψ = res^( Rab given by ρ η = 0, if \H\ is divisible by I, and by ρ η as in (2.1.1), if Η is an /'-subgroup of G. By the isomorphism in (1.2.2) we obtain a corresponding morphism fl'':Q®

Ä

Q ® /?ί

of restriction functors on G, which is explicitly given by G W = W\ Σ (-l)"|Ä r ol[fl r o,resg;(pi i l (resg i (x)))] (4.1.1) 1 ' Ho\/\NH(CT)is

mVn{res^(x)).

an integer, and

Canonical Induction Formulae and the Defect of a Character

41

Note that φη is the character of a projective ΌΗη -module Οψη of rank one, and m Vn ( r e s ^ ( x ) ) is just the number of summands isomorphic to Όψη in a direct sum decomposition of r e s ^ ( M ) into indecomposable OH n -lattices. Let β be a Sylow /-subgroup of Ν//(σ), Η' := QHn (note that Q normalizes σ and hence Hn), and let Ν be an indecomposable summand of r e s I t suffices to show that \Q\ divides the multiplicity/«^(res^Cö)), where θ is the character of Ν, and we may assume that this multiplicity is not zero. Since Ν is a projective G//'-module, Ν is a direct summand of indf'(O) = i n d ^ ( i n d f " ( 0 ) ) . Since H'/Hn = Q is an /-group, Ν is isomorphic to i n d ^ ( L ) for some indecomposable OH„ -lattice L by Green's indecomposability theorem. Mackey's decomposition formula then yields reSg>)=

φ bd*,. seH'/Hn

( » ^ C « )

3

S φ L. seH'/Hn

Since Όψη is a summand of res^'(A^), and since )= 0

0. Definition 1.3. For any ρ-chain C of G we denote by k(C, B, d) the number of characters φ e Irr(/Vc(C)) having defect ά(φ) equal to d and belonging to a ρ-block Β (φ) of NG(C) such that the ρ-block Β(φ)° of G induced by Β (φ) (in the sense of Brauer) is equal to Β. We should remark that the induced block Β(φ)α in the above definition is always defined by a lemma of Knörr and Robinson [KR, 3.2], It is easy to see that the non-negative integer k(C, B, d) depends only on the G-conjugacy class of the p-chain C. So the sum in the following conjecture is well defined. The Ordinary Conjecture 1.4. If Op(G) = 1 and d(B) > 0, then (—1)'C'&(C, B,d) = 0.

Σ

(1.5)

CeX(G)/G

Of course this conjecture is the same as [D2, 6.3]. So it implies Alperin's Weight Conjecture by [D2, 8.3], Furthermore [D2, 7.1] and [D2, 7.3] show that this conjecture can be false when either of its hypotheses O p (G) = 1 or d(B) > 0 fails to hold.

2. The Invariant Conjecture When you try to reduce the above conjecture to the case where G is simple, you quickly discover weaknesses in its formulation. The conjecture as stated says nothing about what happens to the characters it is counting when G is embedded as a normal subgroup in some larger finite group E. But it is exactly this information which is needed to derive the conjecture for Ε from that for G. So we have to strengthen the conjecture in order to carry out the reduction to the simple case. We begin by fixing an epimorphism ε: Ε -» Ε of finite groups with kernel G. So we have an exact sequence 1^ G^ £ A £

1

(2.1)

of finite groups and their homomorphisms. We shall assume that Ε is totally split over J . The conjugation action of Ε on its normal subgroup G induces an action (1.2) of Ε on the ρ-chains C of G. So any such C has a normalizer

48

Ε. C. Dade

NE(C) in E. The image of this normalizer under the epimorphism ε in (2.1) is a subgroup NE(C) =

e(NE(C))

which we may call the "normalizer" of C in E. Since the kernel of the epimorphism ε: NE(C) -» N^(C) is the normal subgroup NG(C) of NE(C), we have an exact sequence 1

Ng(C)^Ne(C)ANe(C)^ 1

(2.2)

of finite groups associated with each ρ-chain C of G. The above group Ng(C) acts by conjugation on the set Irr(/Vc(C)) of all irreducible ^-characters φ of its normal subgroup NG(C). We denote by iVg(C,0) the stabilizer in Ng(C) of any such φ, and by N-E{C,

φ) = ε(ΝΕ(€,

φ))

the image of that stabilizer in E. Since NG(C) is contained in NE(C, φ), the exact sequence (2.2) restricts to an exact sequence 1

WG(C)4Ar£(C,0)AjVg(C,0)--» 1

(2.3)

associated with any ρ-chain C of G and any character φ e Irr(Nc(C)). In addition to the p-block Β of G and the integer d > 0, we now fix a subgroup F of E. Definition 2.4. For any ρ-chain C of G we denote by k(C, B, d, F) the number of characters φ e Irr(/Vc(C)) satisfying ά{φ) = d,

Β{φ)α = Β

and

N^C,

φ) = 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 Σ

(~\)lQk(C,B,d,F)

= 0.

(2.6)

C€£(G)/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 Ε = 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 ρ-chain C of G, the number of irreducible ^-characters ψ of Ne(C) with a given defect lying over a fixed irreducible ^-character φ of Nc(C). Except in rare cases (which, in fact, are not so rare for simple G ) , the subgroup φ ) of Ε by itself does not determine this number. However, the Clifford extension for φ does determine it. So we reformulate the conjecture in terms of Clifford extensions. We start from the exact sequence (2.2) for a ρ-chain C of G. To each φ e ln[Nc(C)) is associated a central extension E[C, φ, 5 ] of the unit group i/(£) of the field £ by the stabilizer A ^ ( C , ) of φ in N^(C). Thus we have a new exact sequence 1-+U(3)^E[C,4>,$]

>Ν-ε{0,Φ)^\

(3.1)

of groups such that U($) is a central subgroup of E[C, φ, J ] · The exact definition of this Clifford extension E[C, φ, such as in [D3, §11], need not bother us here. The important thing is how we can use it to count characters. We denote by Irr( E[C, φ, J ] ) the set of all irreducible projective ^-characters of the extension E[C, φ, J], i.e., of all irreducible ^-characters of E[C, φ, J ] lying over the natural faithful linear ^-character of U ( J ) given by inclusion in J. Then Clifford theory gives us a bijection of Irr( E[C, φ, J ] ) onto the set I r r ( N E ( C ) \ φ) of all irreducible J-characters of NE(C) lying over φ (see [D3, 12.12]). If ψ' e Irr( E[C, φ, £ ] ) is related to ψ e Irr( NE(C) \ φ ) in this way, then their degrees ψ'{I) and ψ(\) satisfy x/r(\) = [NE(C)

:

ΝΕ(€,φ)]φ(1)ψ'(1)

(see [D3, 12.17]). It follows that we can compute the defect of ψ from the defect of φ and the degree of ψ'. Thus the number of ψ with a given defect is determined by the defect of φ and the central extension E[C, φ, without any further knowledge about the structure of NE(C). In fact, we only need to know this central extension to within isomorphisms, i.e., we only need to know the stabilizer Ng(C, φ) and the element a[C, φ, 5 ] of the second cohomology group Η2(Ng(C, φ), U($)) corresponding to the isomorphism class of the central extension E[C, φ,$].

50

Ε. C. Dade

In addition to the ρ-block Β of G, the integer d > 0, and the subgroup F of E, we now fix an element a in the cohomology group H2(F, U($)), where F acts trivially on U(3). Definition 3.2. For any ρ-chain C of G we denote by k(C, B, d, F, a) the number of characters φ e ]IT(NG(C)) satisfying d() = d,

Β(φ)° = Β,

A^(C,0) = F

and

a[C, φ, ff] = a.

As usual, the number k(C, Β, d, F, a) depends only on the G-conjugacy class of the Ρ -chain C. So the sum in the following conjecture is well defined. The Extended Conjecture 3.3. If Op(G) = 1 and d(B) > 0, then Σ

Ce3i(G)/G

(-VlClUC,B,d,F,a)

= 0.

(3.4)

This conjecture implies the Invariant Conjecture 2.5 since the equation (2.6) can be obtained by summing (3.4) over all α e H2(F, i/(#)). If we know that the exact sequence (3.1) must split for all ρ-chains C of G and all irreducible ^-characters φ of NQ(C), then the Extended Conjecture 3.3 is equivalent to the Invariant Conjecture 2.5. This is notably the case when the group Ε ~ E/G is cyclic (see [D3,11.20 and 6.5]). It even occurs when the Sylow r-subgroups of Ε 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 Ε by some central extension £[£] of U($) by E. So we fix an exact sequence 1

U(S) 4 E m

(4.1)

of groups such that U (J) is a central subgroup of £[5]· If Η is any subgroup of E, then //[31 will denote the inverse image of Η in £[5Ί· So / / [ J ] is a

Counting Characters in Blocks, 2.9

51

central extension of £/(3) by Η. We shall assume that 5 is a total splitting field for the extension £[5], in the sense that every irreducible projective ^-character φ of the subextension / / [ 3 ] is absolutely irreducible for every subgroup Η of E. Then the degree 0(1) of any such φ divides the order I //1 of the finite group Η. So φ has a defect ά{φ), the largest integer d > 0 suchthat pd divides the integer \Η\/φ{\). Because the group E[$] is totally split as an extension of U(3), it has a unique subgroup EDR] intersecting U (30 in U (!SH) and having Ε as its image (see [D3, 7.8]). Thus the restriction of 77[J] is the epimorphism τ?[9ΐ] in an exact sequence 1

U m

£[9Ί]

1

(4.2)

making a central extension of U(91) by E. The inverse image H[9\] in E[9\] of any subgroup Η of Ε plays the same role for / / [ J ] as is[9i] plays for . The twisted group order £)[//] of Η over corresponding to the extension is a suborder spanning the twisted group algebra 2t[//] of Η over J corresponding to the extension So we may define the projective p-blocks of H[$] to be the blocks of this unique -order 0 [ / / ] . We denote by Blk(#[ÜK]) the set of all projective p-blocks of H[$]. Any irreducible projective ^-character φ of / / [ J ] is an irreducible character of 21 [H], and hence belongs to a unique projective ρ-block Β (φ) of H[$r]. The natural conjugation action of Η on its twisted group algebra £)[//] over 9Ί has the center Z(Q[H]) as its set of fixed points. So we can use this action to define defect groups D < Ή for any block b e Blk(tf[iK]) (see [D3, 9.3]). The defect d(b) of b is then the non-negative integer such that pd(b) 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 " δ ^ Λ ^ Ρ ) ] " in the latter proposition to the correct "2t[Af G (C)]," it tells us that any projective p-block b of A^G(C)[J], forany ρ-chain C of G, induces a projective p-block of GfSl· As usual, we fix a non-negative integer d. The earlier ρ-block Β of G is now replaced by a fixed projective ρ-block Β [J] of the central extension G[3] of U (3) by G. The projective equivalent of Definition 1.3 is Definition 4.3. For any ρ-chain C of G we define k(C, Β [3], d) to be the number of irreducible projective ^-characters φ of VVG(C)[3] suchthat ά{φ) = d

and

Β(φ)°[*]

= 5 [3]·

52

Ε. C. Dade

The resulting number k(C, Β [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) = 1 and d(B[$]) > 0, then Σ

CeK(G)/G

( - l ) | C | * ( C , ß [ S W ) = 0.

(4.5)

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, 17.15 and 18.5]. Of course it reduces to the Ordinary Conjecture 1.4 when G[31 is a split extension of U($) by G. If C is any ρ-chain of G, then its normalizer NE(C) acts naturally by conjugation on centralizing the subgroup Since NC(C) is a normal subgroup of NE(C), this action leaves invariant, and hence permutes among themselves the characters φ e Ιπ"(Νο(0[3ΐ)· We denote by NE(C, 0, then Σ

(-l)lclk(C,B[d],d,F)

= 0.

(4.8)

CeK(G)/G

Of course this reduces to the Projective Conjecture 4.4 when Ε = G, and to the ordinary Invariant Conjecture 2.5 when is a split extension of U(S) 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 ρ -chain C of G and each character φ e Irr( Nc(C)[$]) is a central extension E[C, φ, 5] of by φ). Our assumption that £ [ 5 ] is totally split over £ implies that E[C, φ, J ] is also totally split over $ (see [D3,11.20]). There is a one to one correspondence between all characters ψ € Ι π · ( Λ ^ £ ( 0 [ 5 ] ) lying over φ and all characters f e Irr( E[C, φ, ). Furthermore, the number of such ψ with a given defect can be computed from the defect of φ and the cohomology class a[C, φ, J ] € H2(N^.(C, φ), U($)) corresponding to the extension E[C, φ, J]· In addition to the above d, β [ £ ] and F, we now fix an element a in the cohomology group H2(F, U (£)). The projective equivalent of Definition 3.2 is Definition 4.9. For any ρ-chain C of G we denote by k(C, B[$],d, F, a) the number of irreducible projective ^-characters φ of jVg(C)[31 suchthat ά{φ) = ά,

5 ( 0 ) G [ 5 ] = ßm,

Nä(C,4>)

= F

and

a [ C , 0 , J ] = a.

The number k(C, 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 Conjecture4.10. If Op(G) then Σ

(—l)'c'&(C,

= 1 and d(B[$])

d, F, a) = 0.

> 0, (4.11)

Ce3i(G)/G

This reduces to the ordinary Extended Conjecture 3.3 when £ [ £ ] is a split extension of U ( J ) by E. It also implies the Invariant Projective Conjecture 4.7, and reduces to that conjecture whenever each a[C, φ, 31 is known to be trivial for all ρ-chains C in G and all φ e Irr(iV G (C)[S]). This last situation occurs in the case where Ε ~ 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 φ e Irr(AfG(C)[5]). The associated Clifford extension E[C, φ, does determine the degrees of

54

Ε. C. Dade

the characters ψ in Irr( A^£(C)[5]) lying over φ, but it does not tell us to which projective ρ-block of any such ψ belongs. We need that extra information in order to compute the projective ρ-block of £[#] induced by Β (ψ). To obtain it we invoke the Clifford theory for blocks developed in [DIL Let b be any projective ρ-block of Ng(C)[SL We denote by Ng(C,b) the stabilizer of b under conjugation by elements of NE(C), and by b) the image NE(C,b)

=

e(NE(C,b))

of that stabilizer in E. In the language of [Dl, 2.17] our present b) would be called N^(C)b. We denote by b) the normal subgroup of Ng(C, b) which would be called N^(C)[b] in the language of [Dl, §2], This notation is chosen because Jacobinski has noticed that the inverse image CE(C,

b) = Ε-1 (CE(C,

b)) η

NE(C)

of Cg(C, b) is precisely the subgroup of all σ e NE(C) which centralize to within inner automorphisms the indecomposable direct summand of the ÜK-order D(WG(C)] corresponding to b. The Clifford extension for the block b and the exact sequence (2.2) is a central extension E[C,b,$\ of £/($) by Cg(C,b). So it appears in an exact sequence 1

_ < Um^E[C,b,$]

_ n[c,b,3] >C-E{C,b)^\

(5.1)

of groups in which U($) is a central subgroup of E[C, b, 5]· In the language o f [ D l , 2.13] the extension E[C,b,$] would be called NE(C)[b]\ There is a natural conjugation action of the group b) as automorphisms of the central extension E[C, b, J ] (see [Dl, 2.19]). Clifford theory for blocks [Dl, 3.7] gives us a one to one correspondence between all projective ρ-blocks Β of NE(C, &)[£] lying over b and all &)-conjugacy classes of projective blocks β' of the central extension E[C, b, of U(J) by Cp{C, b), i.e., of all blocks β' of the twisted group algebra of C^(C, b) over J defined by that extension. Assume that the above block b induces the projective ρ-block of G[£]· The latter block has a normalizer and a centralizer C ^ ß f S I ) in E. It also has a Clifford extension £ [ B [ 5 L £ ] of U($) by and a conjugation action of Λ^(β[3\|) as automorphisms of that Clifford extension. Since b induces 2?[3Ί, its normalizer N^{C,b) is a subgroup

Counting Characters in Blocks, 2.9

55

of A simple extension of the arguments in [Dl, 8.1] shows that Cf (#[#]) is a subgroup of C^(C,b), and hence is a normal subgroup of Ng(C,b). Furthermore [Dl, 8.1] also tells us that the Brauer homomorphism Brc = Br ρ associated with the final ρ-subgroup Ρ = Pn in C sends monomorphically into E[C,b,$] in such a way that the following diagram commutes —

1


,$] of U($) by Ν^(0,φ) associated with any irreducible projective ^-character φ of Ng(C)[$], As in (4.2), this implies the existence of a unique subgroup £[C, 0,91] of E[C, φ, J ] covering φ) and intersecting U($) in U(V\). Factoring E[C, φ, by the kernel 1 + ρ of the natural epimorphism of U(9i) onto U(J), we obtain a central extension E[C,,$] of U($) by N-e{C,φ). We call £[C, ,£] the residual Clifford extension for the character φ. Suppose that the above character φ lies in the projective ρ-block b of Wg(C)[£]. Then Ν^ϋ,φ) is a subgroup of N^(C,b). Furthermore, [Dl, 13.6] tells us that JVg(C,) contains C^{C,b). By [Dl, 13.10] there is a natural monomorphism /z[C, φ] of E[C,b,$] into E[C, φ,$] such that the following diagram commutes 1

U(d)

E[C, b, m

i?[CAM>

Cf(C,b)

1 (5.3)

ß[C, φ] 1

υ (3)

E[C,4>,$]

η

•

ΝΒ&,Φ)

—•

1

and the conjugation actions of Nj?(C, φ) on E[C, b, J ] and on E[C, φ, J ] are preserved. When the block b containing φ induces we may compose the monomorphisms in the commutative diagrams (5.2) and (5.3) to obtain a mono-

56

Ε. C. Dade

morphism

μ[Β[$], C, φ] = μ[ϋ, φ] ο Brc: Ε[Β[3], ff] ~ E[C, φ, £] of groups such that the following diagram commutes 1

—

1

and the conjugation actions of Ν^ϋ,φ) on 31 and E[C, φ, J ] are preserved. If Ψ is an irreducible projective ^-character of NE(C) lying over φ, then we can compute the projective ρ-block B ( f ) E m of E[$] by the following recipe: Let ψ' be the irreducible projective character of E[C, φ, J ] corresponding to ψ under Clifford theory for φ. Then ψ' belongs to a projective ρ-block Β(ψ') of E[C, 0 , 3 ] . This projective ρ -block corresponds to a unique projective block Β (ψ') of the residual Clifford extension E[C, 0 , 3 ] · The latter block lies over a unique 0)-conjugacy class of projective blocks β of the restriction of E[C, φ, 3] to an extension by the normal subgroup of Νp(C, φ), i.e., to the image of £ [ ß [ 3 ] , 3 ] under the monomorphism μ[Β[$], C, φ] in (5.4). Each such β is the image of some unique projective block β' of £ [ £ [ 5 1 , 5 ] under that monomorphism. The resulting ß ' form a single Ng(C, 0)-conjugacy class, which is contained in a unique TV^. ( 5 [3D -conjugacy class. The projective ρ-block Β ( ψ ) Ε ^ of E[$] corresponds to that Λ^ ( ß [3]) -conjugacy class under Clifford theory for the block Β [ff]. We assume from now on that our fixed subgroup F of Ε satisfies

CE(Bm

F

1,

in which is a central subgroup of F[3], Because this central extension is totally split over there is an associated residual central extension F [ £ ] of U($) by F , with related exact sequence 1

U($)

F[£]

• F -> 1 .

The second element in our ordered pair is a monomorphism κ of the Clifford extension -Et-Bfö], 31 for the block B[$] into the central extension F[31· So it occurs in a commutative diagram -

1

—•

ί/(ϊ)




< -

1

—•

U(D)

-

1 (5.6)

-

Ftf]

•

F

—•

1

with exact rows. Furthermore, this monomorphism κ: J ] >—> F[51 carries the conjugation action of F on the former group (the restriction of the conjugation action of Λ ^ ( £ [ £ ] ) on that group) into the conjugation action of F on F [ £ ] . If (F\$\',K') is also an augmented central extension of U(£) by F, then an isomorphism of ( F ^ L K ) onto ( F f g ] ' , A:') is an isomorphism γ of F [ # ] onto F [ 5 ] ' as extensions of U(5) by F such that the induced isomorphism γ of F [ £ ] onto F[3T satisfies

From now on we fix, in addition to the earlier d, mented central extension (F[31>O of by F .

and F , an aug-

Definition 5.7. Forany ρ -chain C of G, we denote by k(C, B[$],d, F [#],«•) the number of irreducible projective ^-characters φ of //G(C)[31 suchthat ά(φ) = ά,

Β(φ)°Μ

= Β[$],

and the augmented central extension ( E [ C , φ, 3Ί> μ[·β[3Ί» C, φ]) of

U($)

by Ng(C, φ) is isomorphic to the augmented central extension (F[51, κ) of U(-8) by F (so that, in particular,

is equal to F ) .

58

Ε. C. Dade

As always, the number k(C, ß[31> d, F[31, κ) depends only on the G-conjugacy class of C. So the sum in the following conjecture is well defined. The Inductive Conjecture 5.8.

Σ

If Op(G) = 1 and d(B[$]) > 0, then

(~l)lQk(C,B[$),d, ?[$],*) = 0.

(5.9)

CeX(G)/G

Of course this conjecture implies all our previous ones. With a great amount of work it can be shown to hold for all finite groups if it holds whenever G is a non-abelian finite simple group. So it can be proven by checking it for all such G.

6. Simple G The verification of the Inductive Conjecture 5.8 for a given non-abelian finite simple group G requires only a finite calculation. The isomorphism classes of central extensions G[3] of U (3D by G correspond one to one to the elements of the Schur multiplier H2(G, U(3D), which is a finite group of known structure for each simple G (see [Atlas]). So there are only a finite number of choices for G[31 once G is given. If G[J] is fixed, there are an infinite number of possible choices for its extension group £[#]. However, the inductive conjecture can be shown to hold for all these choices provided it holds for one of them satisfying the extra condition that conjugation by suitable elements of £"[31 induces every possible automorphism of the central extension G[#] of U(5). Such an E[$] always exists. So we only need consider one £[5] for any given G ^ ] . Every term in the sum on the left side of (5.9) is zero when the integer d is so big that pd does not divide |G|. Hence the inductive conjecture holds trivially except for a finite number of choices of d. There are, in any case, only a finite number of choices for the other parameters F, F[3r] and κ appearing in that conjecture. So the verification of it is a finite calculation for any fixed G. The Inductive Conjecture 5.8 has been verified completely for many finite simple groups G, including all the Mathieu groups, thefirstthree Janko groups, the Suzuki groups Sz(q) and the linear groups PSLiiq). Partial verifications have been made for the two classes of Ree groups, the groups Gziq), and the linear groups PSLj, (q). In all these cases the inductive conjecture is equivalent to weaker conjectures, such as the Invariant Projective Conjecture 4.7 or the Extended Projective Conjecture 4.10. For example, if every Sylow r -subgroup

Counting Characters in Blocks, 2.9

59

of Ε is cyclic, for every prime r, then the inductive conjecture is equivalent to the invariant projective conjecture. This is enough to handle the cases where G is a sporadic simple group or an alternating group An with η Φ 6. It even covers many cases where G has Lie type. The inductive conjecture reduces to the extended projective conjecture when the subgroup is trivial. Using [Dl, 0.3b] we can show that this happens for all groups of Lie type when ρ is their defining characteristic. Another situation where the inductive conjecture reduces to the extended projective one occurs when Ε is abelian. This is what enables us to handle PSL2(q) completely. Note added in proof. Since this paper was written, the inductive form of the conjecture has been verified completely for more simple groups, including the McLaughlin group, the Held group, the third Conway group, the Tits group, and the Higman-Sims group. There have also been partial verifications for a few more families of groups of Lie type. So far no counterexample has been found, although there was a brief flurry of interest on the Internet in January and February, 1997, when a counterexample was erroneously thought to exist.

References [Atlas] J. Conway, R. Curtis, S. Norton, R. Parker and R. Wilson, Atlas of Finite Groups, Oxford University Press 1985 [Dl]

E. Dade, Block Extensions, 111. J. Math. 17 (1973), 198-272.

[D2] [D3]

E. Dade, Counting Characters in Blocks, I, Invent. Math. 109 (1992), 187-210. E. Dade, Counting Characters in Blocks, II, J. Reine Angew. Math. 448 (1994), 97-190.

[D4]

E. Dade, Counting Characters in Blocks, III, in preparation.

[KR]

R. Knörr and G. Robinson, Some Remarks on a Conjecture of Alperin, J. London Math. Soc. (2) 39 (1989), 48-60.

[RS]

G. Robinson and R. Staszewski, More on Alperin's Conjecture, Asterisque 181/182(1990), 237-255.

Department of Mathematics The University of Illinois at Urbana-Champaign 1409 W . G r e e n St. Urbana, IL 61801, USA Email: [email protected]

The Defect Groups of a Clique H.

Ellers

Alperin's Weight Conjecture and Brauer's First Main Theorem on Blocks are strikingly similar. Let k be an algebraically closed field of characteristic p. For any finite group G and any ρ-subgroup β of G, let a(G) be the number of irreducible &G-modules and let ÜQ{G) be the number of irreducible kG-modules with vertex Q\ let b{G) be the number of blocks of kG and let frß(G) be the number of blocks with defect group Q. It follows from Brauer's First Main Theorem that b{G) =

YibQ{NG{Q))t Q

where Q runs through a set of representatives for the conjugacy classes of ρ-subgroups of G. Alperin's Conjecture suggests that a(G) =

Y^aQ(NG(Q)), Q

where Q runs through the same set. Are there other formulas of this type? This is a report on work that shows that when G is ρ-solvable, there is a whole family of such formulas, one for each normal subgroup Η of G, with Alperin's Conjecture as the case Η = 1 and Brauer's First Main Theorem as the case Η = G. Let Η be a normal subgroup of G. Let kGH be the centralizer in kG of kH. The group G acts on kGH by conjugation, so there is also a natural conjugation action of G on the set of isomorphism types of irreducible kGH-modules. By Theorem 2.5 in [Ell], there is an analog of Clifford's Theorem for the restriction of an irreducible kG-module V to kGH: the restriction VKGH is semi-simple, the set of all isomorphism types of simple summands of VKGH is a G-conjugacy class, and each isomorphism type of simple summand occurs the same number of times in a decomposition of VkGH. We say irreducible kG -modules V and W are Η-equivalent if there is a nonzero submodule of the restriction VkGH that is isomorphic to a submodule of the

62

Η. Ellers

restriction WkGH. (It follows from the analog of Clifford's Theorem that this is an equivalence relation. If there is one irreducible submodule of VkGH that is isomorphic to a submodule of WkGH, then every irreducible submodule of VkGH is isomorphic to a submodule of WkGH and every irreducible submodule of WkGH is isomorphic to a submodule of VkGH.) We call the equivalence classes H-cliques. The partition of the set of irreducible kG -modules into //-cliques is similar to the partition into blocks. When Η = G, irreducible kG-modules belong to the same //-clique if and only if they belong to the same block. In general, when Η is a proper subgroup, the partition into //-cliques is a refinement of the partition into blocks. When Η = 1, irreducible kG -modules belong to the same //-clique if and only if they are isomorphic. The papers [Ell], [E12], and [E13] develop a theory of cliques as similar as possible to Brauer's theory of blocks. In particular, defect groups of a clique are defined in such a way that, for ρ-solvable groups G, there is a First Main Theorem: for any ρ-subgroup β of a ρ-solvable group G, the number of //-cliques with defect group Q of irreducible kG -modules is equal to the number of NH( Q) -cliques with defect group Q of irreducible kNG(Q)-modules. Now we explain how defect groups are defined. We need the following theorem of Green [Gr]. To state Green's theorem, we need some definitions. Definition 1. A G-algebra is a finite dimensional algebra over k on which the elements of G act as algebra automorphisms. If A is a G-algebra and Κ is a subgroup of G, then AK is defined to be {α ς. A \ ax = a for all χ € /Π. For any a e Ακ, (a) = Σί agi - where {#,} is a set of representatives for the right cosets Kgi of Κ in G. Theorem 2. Let G be a finite group, let A be a G -algebra, and let Μ be a maximal ideal of A°. Then the the set of all subgroups D of G minimal (with respect to inclusion) with the property Τβ(Α°) £M.isa G-conjugacy class of ρ-subgroups of G. For any maximal ideal Μ as in Green's theorem, the groups D are called the defect groups of M. Consider the classical case A = kG. For any centrally primitive idempotent e of kG, there is a unique maximal ideal Μ of kG° with e & M. The defect groups of Μ are the same as the defect groups in Brauer's sense of e. If V is an indecomposable kG -module, then Endfc(V) is a G-algebra and End*(y) G is a local ring. The defect groups of the unique maximal ideal of End^(V) c are the vertices of V.

The Defect Groups of a Clique

63

Let G bean Η -clique of irreducible kG -modules. We wish to use Green's Theorem to define defect groups of G. To do this, we need to find an appropriate G-algebra and an appropriate maximal ideal of its subalgebra of G-fixed elements. First, we will examine a G-algebra which is not quite the correct one to use, but which is closely related to the correct one. Note that if V and W are irreducible -modules in the same Η-clique, then annkGH(V) = ann

kGH(W).

Definition 3. Let Η be a normal subgroup of the finite group G. Let C bean Η -clique of irreducible fcG-modules. Let Ve be an irreducible fcG-module in 6. (1)

AGM(e)

=

kG»/aankGH(Ve),

When ambiguity is not possible, we will drop the subscript G, Η from the notation and write simply (2) For any a e kGH,

ä is its natural image in

The algebra kGH/J(kGH) is semi-simple, with one simple summand for each isomorphism type of irreducible kGH-module; the action of G permutes these summands. The G-algebra A{&) is naturally isomorphic to the G-subalgebra of kGH/ J(kGH) consisting of all the simple direct summands corresponding to modules in the single G-orbit of irreducible kGH-modules associated to e. Whenever it is convenient, we will identify .A(C) with this G-subalgebra of kGH/J(kGH). Using the analog of Clifford's TheoH rem, we see t h a t k G Π J{kG) — J{kGH)\ it follows that the natural map kG" kG/J(kG) induces an injective map kGH/J(kGH) kG/J(kG)\ therefore A(Q)° is contained in the center of J4(C). Since ~4(C) is the direct sum of a single G-orbit of simple ^-algebras, we get the following lemma. Lemma 4. Let G, Η and C be as in Definition 3. Then A(C) G = k. The algebra yi(G) is the most natural G-algebra to associate to S. However, thinking about the classical situation G = Η shows that it is not the correct algebra to use to define defect groups. When G = Η , is 1 -dimensional and the action of G is trivial, so the defect groups of the unique maximal ideal 0 of A(Q)° are the Sylow /^-subgroups of G, not the defect groups of the block identified with C. We need a larger algebra and an ideal that together encode information about four things: Λ(6), the action of G on the action of G on kG, and the natural map kGH A(Q). The required algebra is provided by the construction of Definition 5.

64

Η. Ellers

Definition 5. Let G be a finite group and let Λ be a G-algebra. Then A*kG is the following algebra. As a vector space, A * kG is A a * χ = a ® x . Multiplication is defined by

kG with

X^ (αϊ * Χΐ)(α2 * Xl) = a\Ü2 * X\X2 whenever a\ and A(C) * kG. Now we must identify an appropriate maximal ideal of (Λ(6) * kG)G. First, we need to find the //-fixed elements of Λ(6). This is similar to the usual determination of the center of a group algebra. Think of elements of the skew group ring as linear combinations with coefficients in A(G) of elements in G. Since the action of Η is trivial on the coefficients, the coefficients of any Η - fixed element must be constant on //-conjugacy classes in G. Since the class sums for //-conjugacy classes form a basis for kGH, we obtain the following lemma. Lemma 6. With the above notation, (Λ(6) * kG)H = A(£) *

(kGH).

The appropriate maximal ideal of (Λ(6) * kG)° is provided by the intersection of (Λ(6) * k G ) ° with the kernel of the following map. Definition 7. Let G, H, and G be as in Definition 3 Ac.ff.e :

* (kGH)

A(G)

is the map given by Ac,H,e(a *b) = ab for all a € Λ(6) and all b € kGH. Whenever ambiguity is not possible, we will drop the subscript G, H, G from the notation and write simply A. It is easily checked that the map A is a surjective G-algebra homomorphism.

The Defect Groups of a Clique

65

In the special case Η = G, the map Λ is a very familiar object; in this case C is the set of irreducible modules in a block Β of kG, J4(C) is a one-dimensional G-algebra with trivial action of G, A(Q)*kG is the group algebra kG, Λ(6) * (kGH) is the center of the group algebra, and A is the central character of kG corresponding to the block B. Since ( Λ ( 6 ) ) σ is one-dimensional, and since A is a G-algebra homomorphism, it follows that (.A(C) * kG)° Π Ker(A) is a maximal ideal of (.A(Q)*kG)G. Definition 8. Let Η be a normal subgroup of the finite group G. Let C be an //-clique of irreducible &G-modules. Let A be the map of Definition 7. The defect groups of C are the defect groups of the maximal ideal (.A(C) * kG)° Π Ker(A) of (A(G) * kG)°. It is immediate that in the case Η = G, this agrees with the usual definition of the defect groups of a block. It is also easy to check that when Η = 1 , there is just one irreducible module in the clique C and the defect groups are just the vertices of that module. A warning is needed about the possible dependence of defect groups on Η . It is sometimes possible to change Η without changing the set of modules C. (For example this happens if Η is replaced by a normal subgroup H\ of G with H\ c Η and H/H\ a group of order prime to p.) However, the algebras and maps in the definition of defect groups depend not only on the set C but also on Η . Thus defect groups are, at least in principle, only defined relative to Η . It should always be clear from the context which Η is intended. Whether changing Η can change the defect groups in a case when it does not change G is not known. It is natural to speculate, by analogy with blocks, that the defect groups only depend on the vertices of the modules in the clique; however very little is at present known about this. On the positive side, if D is a defect group of the //-clique C of irreducible &G-modules, and if V is any module in G, then V is D-projective. With this definition of defect groups of a clique, the main theorem of [E13] is the following. Theorem 9. Let k be an algebraically closed field of characteristic p, be a p-solvable finite group, let Η be a normal subgroup of G, and let a ρ-subgroup of G. Then the number of Η-cliques with defect group irreducible kG-modules is equal to the number of ΝH (D)-cliques with group D of irreducible kNo (D)-modules.

let G D be D of defect

66

Η. Ellers

For any normal subgroup Η and any ρ-subgroup Q of a finite group G, let c//(G) be the number of //-cliques of irreducible ZcG-modules and let CH,Q{G) be the number of //-cliques with defect group Q of irreducible £G-modules. In terms of the notation of the first paragraph, a(G) = ci(G), aQ(G)

= c i , e ( G ) , b(G)

= Cg(G),

and bQ(G)

= Cg,Q(G).

Combining all

possible defect groups, we obtain the following corollary. Corollary 10. With the above notation, if G is ρ-solvable, then

ß

where Q runs through a set of representatives for the conjugacy classes of ρ-subgroups of G. There are groups G that are not ρ-solvable for which the conclusion of Theorem 9 is false, even in the familiar case Η = 1. However, the corollary (which is Alperin's Conjecture when Η = 1) may turn out to be true for all groups G.

References [Al]

J. L. Alperin, Weights for finite groups, in: The Areata conference on representations of finite groups (P. Fong, ed.), Proc. Symp. Pure Math. 47, Part 1, Providence, R.I., 1986, 369-379.

[Gr]

J. A. Green, Some remarks on defect groups, Math. Z. 107 (1968), 133-150.

[Ell]

H. Ellers, Cliques of irreducible representations of p-solvable groups and a theorem analogous to Alperin's conjecture, Math. Z. 217 (1994), 607-634.

[E12]

H. Ellers, Cliques of irreducible representations, quotient groups, and Brauer's theorems on blocks, Canadian J. Math. 47(5) (1995), 929-945.

[E13]

H. Ellers, The defect groups of a clique, p-solvable groups, and Alperin's Conjecture, J. Reine Angew. Math. 468 (1995), 1 ^ 8 .

Department of Mathematics Northern Illinois University DeKalb, IL 60115 Email: [email protected]

Representations of GLn(K) and Symmetric Groups Karin

Erdmann

Let Κ be an infinite field of characteristic ρ > 0. It is known for some time that there is a close connection between the representation theory of the symmetric groups over Κ and the theory of polynomial representations of GLn(K). For Κ — C there is already the work of I. Schur [S], and later one has [CL], [G], [J] and others. More recently Cline, Parshall and Scott defined quasi-hereditary algebras in order to deal with highest weight categories as they arise in the representation theory of semisimple complex Lie algebras and algebraic groups [CPS1]. Subsequently, the structure of quasi- hereditary algebras in general was studied in [DR], [R] and by others. This led to the discovery of a new class of modules which are parametrized by highest weights, called canonical modules, or 'tilting modules' [R]. For the case of the highest weight category associated to GLn{K), these modules provide a new connection to the representation theory of the symmetric groups and give new insight to longstanding problems, such as decomposition numbers and dimensions of simple modules. The aim of this paper is to give an exposition of relevant properties of the canonical modules for GLn(K)\ in particular we shall explain the relationship to symmetric groups and give an overview of some recent results on the problems of decomposition numbers and dimensions of simple modules.

1. Symmetric Groups Sr and Schur Algebras 1.1. We consider modules of the group algebra of the symmetric group Sr over K. For each partition μ of r, let S ß denote the Specht module corresponding to μ; it is defined characteristic-free. If char A' = 0 then S ß is simple and this gives a full set of pairwise non-isomorphic simple modules. Over characteristic p, the simple modules for ΚSr are labelled as Dx where λ runs through the p-regular partitions of r. If λ is p-regular then the Specht module has unique simple quotient Ζ)λ ; and all other composition factors

68

Karin Erdmann

of SK are of the form ΰμ for λ < μ. Here < denotes the dominance order of partitions. The decomposition numbers can be taken as the composition multiplicities of the Specht modules 5 μ ; we shall use the notation [SM : Dx], We shall need the following sets labelling simple modules. First, Λ + ( n , r) is the set of partitions of r with at most η parts, and Λ+(n, r) denotes the subset of ρ -regular partitions.Moreover, we write Λ + ( n ) for the set of all partitions with at most η parts, and A+(n) is the subset of ρ-regular partitions. We also write |λ| = r if λ is a partition of r. Moreover, we denote by A(n,r) the set of unordered partitions of r with at most η parts. For each λ e A ( n , r ) let Μ λ be the permutation module corresponding to λ. If λ is a partition then 5 λ is eplicitly defined as a submodule of Μ λ . It has strong uniqueness properties which guarantee that there is a unique indecomposable summand of Μ λ containing 5 λ ; and this is by definition the Young module corresponding to λ, denoted by Υλ. The modules Y x have filiations by Specht modules and also by duals of Specht modules. 1.2. [G,M] Let Ε be a fixed η-dimensional vector space over Κ. The symmetric group Sr of degree r acts on the right of E®r by place permutations. The classical Schur algebra can be defined as the endomorphism ring of this module, S(n,r)

·.= End S r (E® r )

As a module over ΚSr, E®r is isomorphic to the direct sum φ Μ λ where the sum is taken over A ( n , r ) . Incase η > r, the Schur functor / from the category of S(n, r)— modules to the category of Κ S r - modules is an important tool; we recall the definition. Let e : E®r E®r be the projection with image M(l ^ which is zero on Μ λ for λ φ (V), then e is an idempotentin S(n, r), and one sets f{M) := eM. The algebra eS(n,r)e is isomorphic to the group algebra of S r . Namely, eS(n, r)e is isomorphic to the endomorphism ring of M ( i r ) but Af ( i r ) is isomorphic to Κ Sr. Hence one can view fM as a left module for KSr. In particular, the above argument also shows that f(E®r) = Κ§r.

Representations of GLn(K)

and Symmetric Groups

69

2. GL„ (tf)-Modules and Tilting Modules 2.1. Let Κ be an infinite field. Fix an integer η > 1, let G be the group GLn(K). We denote by M r the category of finite- dimensional polynomial representations of G which are homogeneous of degree r . Most important is the natural «-dimensional G-module which we denote by E . Then Ε belongs to M i and for any r > 1, the r-fold tensor product E®r is in Mr, and also r-fold symmetric powers and r-fold exterior powers, for r < n. The category Mr is equivalent to the category of S(n, r)-modules (see [G, M]). The simple modules are parametrized by highest weights as L(k), for λ e A+(n,r). 2.2. For λ € A + ( n , r) let Δ (λ) be the Weyl module with highest weight λ, and let ν ( λ ) be its contravariant dual (see e.g [G] or [M] for an explicit construction). For example, the r-fold symmetric power Sr(E) is isomorphic to V(r). Moreover the r-fold exterior power Ar(E) is isomorphic to V ( l r ) for 1 < r < n. Actually this is simple, = L ( ( l r ) ) and isomorphic to its dual. In general, the important property is that Δ (λ) has a unique simple quotient isomorphic to L(A); and all other composition factors are of the form Ζ,(μ) for μ < λ; here < is dominance order of partitions. This is analogous to the property of Specht modules Sx for p— regular λ as mentioned above; except that the order is reversed. The category M r is a highest weight category in the sense of [CPS1] where the weight poset (A, 2n — 2; in particular always for η = 2. We shall sketch the first part of the proof. The Weyl module Δ ( ( ρ — l)p) (the Steinberg module) is known to be simple, hence it belongs to Τ (see 2.5.1.) It follows that Τ (γ) ® Δ ( ( ρ - 1)ρ) also belongs to Τ and it has Τ (γ) as a direct summand since γ is its highest weight. So the first part will follow if one shows that Τ (γ) ® Δ ( ( ρ — 1)ρ) Τ{τ)ρ belongs to Τ, and for this it suffices to establish that Δ ( ( ρ — l)p) ® T{x) F has a Δ-filtration and a V-filtration. This is now a consequence of a result proved by Jantzen [Ja,II.3.19]. Namely one has v ( ( p - D p ® v ( / i ) F = V((p - i ) p + Δ ( ( ρ - 1)Ρ) Θ Δ ( μ ) ^ ^ Δ ( ( ρ - 1 )p +

ρμ),

ρμ).

For the last part and a related conjecture, see the discussion in [D2] after 2.1. Example. Assume η — 2 and work with SLjiK). Then using 2.6, the partition λ = (λι, λ 2 ) corresponds to the weight m = λι —

72

Karin Erdmann

(a) Suppose 0 < m < ρ — 1, then T(m) = A(m) since in this case Δ(m) = L(m) = V(m); this is well-known (and easy to see by direct calculation, in the m-fold symmetric power.) (b) If m > ρ write m = kp + i where 0 < i < ρ — 1. Then ^ \ Km) -

T ( m

T{k)F®T{p_ 0

1) T

(p

+

i = p- 1 ,·) i + i) where i + j = ρ — 2 (see [X]). In particular this gives the Δ-quotients for T(p + i).

3. Symmetric Groups and Schur Algebras 3.1. We have seen that the r-fold tensor product is a direct sum of canonical modules. More precisely, Theorem [D; IV], There is an isomorphism of S(n, r) -modules E®r £ (&χάχΤ(λ) where the sum is taken over Λ+(n, r). Moreover the multiplicity dx is equal to the dimension of the simple module Dx of the symmetric group S r . We give an outline of the proof, for the details see [El, 4.2]. Let E®r = ΦάλΤ(λ), where the sum is taken over A + (n, r), we have to find the multiplicities. One reduces to the case η > r, then the Schur functor / : M r — KSr - mod is defined. One has f(E®r) = KSr and / ( Γ ( λ ) ) = Υλ' Ka where Υλ is the Young module corresponding to λ' and Ka is the alternating representation (see [D2;3.8]). It is known that Κλ ® Ka is projective if and only if λ is p-regular, and if so then it is isomorphic to P(Dk). The multiplicity of P{DX) as a direct summand of Κ§r is dim£>\ 3.2. If A is any quasi-hereditary algebra with respect to (A, λ in the same block as λ are ρ-regular. Then βΔ5'(λ) = As>(S) and eTy(X) = TS>(X). The first isomorphism is clear. For the next one, recall that S(n, r) is the endomorphism ring of the KSr -module E®r, hence there is a canonical map Pn • K$r -> End5(£'®r) . It follows from [CP] that pn is surjective. For the other parts we refer to [El, 4.2], This may explain the analogy between module structure of Specht modules and of Weyl modules mentioned above; and also the filtration properties of Young modules as compared with those of canonical modules. In fact, there is now a generalization. Suppose R is any finite-dimensional algebra, Τ an R -module and A = E n d ^ ( r ) . In [CPS2], the relationship between the representation theories of R and A is studied systematically, especially for A quasi-hereditary. This work includes amongst others an abstract 'Specht/Weyl module correspondence'. 3.2.1.

3.3. We shall now see that the above result explains why decomposition numbers for symmetric groups are related to the quasi- hereditary structure of the Schur algebras. Theorem. Let λ, μ e A + ( n , r ) , then [Γ(λ) : Δ(μ)] = [Δ(μ') : L(k')]

74

Karin Erdmann

If in addition λ is p-regular then also [Τ (λ) : Δ (μ)] = [S* : 0 λ ] , The first part is due to Donkin (in the case when λ is p-regular it can also be deduced from the second part.) For the second part, let S' be the Ringel dual of 5. By [R] there is an equivalence

which takes the indecomposable injective module β' 5 (λ) to Τ (λ) and also ν£(μ) to Δ (μ). Consequently [Γ(λ) : Δ(μ)] = [β' 5 (λ) : V' s (ji)]. By 'Brauer-Humphreys reciprocity' (see [CPS1], the second multiplicity is equal to [Δ' 5 (μ) : Ζ/5(λ)], and by the identification in 3.3. it is equal to [Sß : Ζ)λ]. Remark. (1) In cases when the decomposition numbers for S r are known this may be used to find the Δ-filtration of Γ (λ) for λ p-regular. The column of Dx in the decomposition matrix for partitions of at most η parts gives the δ- quotients of Τ (λ). (2) By 2.6, the filtration multiplicities [Τ(λ) : Δ(μ)] depend only on the equivalence class of λ, μ with respect to that is only on the restriction to SLn(K). In particular it follows that the decomposition numbers [Sß : D x ] also only depend on the equivalence class. 3.4. Suppose all λ e A+(n, r) are p-regular. Then it follows from 3.2 (and using Ringel's equivalence) that the category of K S r / I n -modules which have filiations by dual Specht modules (5 λ )* for λ e A+(n, r) is contravariantly equivalent to the category (Δ) of 5-modules with Weyl filtration. Correction. Some statements in the introduction of [El; p. 124] do not agree with what is proved. (1) Equivalence between categories of S(n, r)-modules with Δ-filtration, all Δ having p-regular partitions as weights, and the corresponding categories of modules with Specht filtration. It should have said contravariant equivalence (as it follows from 4.3(b)). (2) The category of modules with Specht filtration. Proofs are given for the category of KSr/In but not of Κ S r ; this should have been said in the introduction. The same in 4.4, the second Theorem. Actually, when ρ > 2 the more general statement with KS r (instead of the quotient algebra) is true; it is proved in [CPS2]; see also [D3, section 4.7]. For ρ = 2 it is false.

Representations of GLn(K)

and Symmetric Groups

75

4. Decomposition Numbers 4.1. One of the basic problems in modular representation theory is that of understanding the decomposition numbers. In the case of symmetric groups these are the composition multiplicities [Sß : Dx] of the Specht modules. There is a similar problem for representations of GLn(K), namely the problem of understanding composition multiplicities of the Weyl modules [Δ(μ) : Ζ,(λ)]. This is considered to be difficult; and an answer is suggested by the Lusztig conjecture for type A. The following shows that both problems are the same: Theorem. Let η be fixed. Then the composition multiplicities {[Δ(μ) : £(λ)], λ G A + (/j)} are the same as the decomposition numbers {[S^ : Dx] : λ G Λ+(n), μ e A+(«)} if r varies. It was proved by James [J3 ] that the decomposition numbers for S„ are composition multiplicites of Weyl modules, but the converse does not follow if one takes η = r. We will explain the proof of the Theorem. First, composition factors of Weyl modules imply decomposition numbers, this follows directly from 3.4 if one specialises to λ ρ-regular. To show that decomposition numbers for Sr imply composition factors of Weyl modules, let 6 be the partition (n — 1, η — 2 , . . . , 2,1, 0) with η parts. For a partition λ of r let ί(λ) — ρλ + (p — 1)5, this is a partition of t(r) := pr + (£)(ρ - 1). We use [Ja, II.3.19] (see also 2.7), and a special case of 2.6 and have A(/x) f ® A((p - 1)δ) = Δ(ί(μ)),

T(X)F ® A((p - 1)5) = Γ(ί(λ)).

Consequently we get [ 7 W ) ) : Δ(ί(μ))] = [Γ(λ) : Δ(μ)] The important fact is that t(k) is always p-regular. Applying 3.4 shows that [Stß

and the statement is proved.

. flrtj

= [Δ(μ/}

.

L(A/)]

76

Karin E r d m a n n

4.2. One will conclude that the problem of understanding decomposition numbers is in fact hard. Also, bearing in mind the Lusztig conjecture, one may not expect to find easy closed formulae. Nevertheless there may be interesting recursive characterizations. 4.3. For the case when η = 2, G. D. James discovered an algorithm for the decomposition numbers [Jal, 2], There is a short proof due to Donkin, using the results on twisted tensor products of canonical modules (see [El]). It is a consequence of the properties we have listed in 2.8; we will now describe a variation of the algorithm. We work with SL2(K) and we shall use 2.8. Recall (see 3.3.1) that for a partition with two parts, [Sß •• Dk]

=

[T(s)

:

A(m)]

if s — λ\ — λ 2 and m = μ\ — μι. We define therefore the generating polynomial corresponding to the column for Dk in the decomposition matrix by f

s

{z) =

Y

m

d

J

m s Z

m> 0 ß

x

where dms = [S : D ], as above. The known properties of Specht modules translate into dss - 1 and dms φ 0 implies m < s and m = s (mod 2). Moreover the facts on Τ(s) listed in 2.8 translate directly in the following. 4.3.1.

(a) For 0 < s < ρ — 1 we have fs(z) = z-5.

(b) fk P + (P-i)(z) = z p - l f k ( z P ) . Moreover

f

k p

(c) We have

f

+i(z) = p + i

(z)

f

=

P

+i(z)fk-i(z

zp+i

p

for 0
• there is only one ÖG-block b which covers b [Al]. (We say that an ÖG-block b covers b if bOH is ίβοηιοφΜϋ to a direct summand of bOGHxH as Ό(Η χ Η)-modules.) The four definitions are not equivalent, but if any two of the four types of block induction are defined, they are the same. In general we have bG

= b = *

b ^

G

fcextG

= b = ^

-

b

and U

G )

= b = * b

e x t G

=

b.

There are examples showing that it is not true in general to reverse any of the above arrows or get any further implications among these four definitions. [B12]

2. Strong Covering and its Characterizations In order to investigate extended block induction more closely, we would like to define extended induction in a similar way to the definition of Alperin-Burry induction. Hence we introduce the definition of strong covering of blocks which is indeed a strong case of covering of blocks. (See the remark following Proposition 2.2.) Definition 2.1. Let G be a finite group and Η a subgroup of G. Let b be an 0/7-block and b a central idempotent of OG. Then we say b strongly covers b if λΒ ο ( B r G ) ( b ) φ 0.

On Extended Block Induction

87

By definition, we can see that b induces to a block b of G in the extended sense if and only if b is the unique block which strongly covers b. Also for a given block b of Η there is at least a block b of G strongly covering b since ο (Brg)(l) = 1 ^ 0 . The following properties are useful when we study strong covering. (See also Prop 1.4 in [Wh]) Proposition 2.2. Let Η be a subgroup of G, b and b be OG, resp. Then the following are equivalent.

OH-blocks

(1) b strongly covers b. (2) S(Brg)(fc) i

J(ZOH).

(3) Let f : bOH —• (bOG)HxH be the 0[H χ H]-homomorphismdefined by f ( x ) = bx and g : (bOG)HxH —• bOH be defined by g(y) = bBr^jiy); then g ο f is an automorphism of bOH. (4) Let θ e Irr(ü). Then θβ(1)ρ

=

eb(l)p.

Remark. The equivalence of (1) and (3) shows that strong covering implies general covering. So Alperin-Burry induction implies extended induction. The equivalence of (1) and (4) is useful when we compute examples. H. Ellers, G. Hill and Y. Fan have given characterizations of Alperin-Burry induction in terms of ρ-local subgroups [El]. The following theorem, part of which is analogous to Fan's theorem, gives some ρ -local characterizations for extended induction. Theorem 2.3. Let b be a block of FH with defect group D and b a central idempotentof FG. Denote Β := Br^ w ( D ) (£) = Σί h and Β := B r c c ( D ) 0 ) , where b, 's are blocks in FDCH(D). (By Brauer's First Main Theorem, Β is a block of FNG(D). ) Then the following are equivalent. (i)

b strongly covers b.

(ii)

Brg(&)y = b for some y e

(iii)

Β Brg ( o)(B)z = Β for some Ζ e

(iv)

Β in NG(D)

(ν)

Β in DCg(D)

b(FH)".

(

(or Ch(D)

strongly covers Β in (or CG(D))

resp.).

FCH{D)N"(D). NH(D).

strongly covers

bi, for all i, in

DCH(D)

88

(vi)

G. Huang Β in DCG(D) (or CQ(D)) strongly covers B,, for some i in (or Ch(D) resp.).

(vii) There exists Χ e FCH{D)Ch{D)

DCH(D)

such that Β B r g ^ j C ß ) * = Β.

(viii) For any i, there is a Zi € FCH(D)Ch(D)

suchthat bt B r ^ J ) (B)zi =

h. (ix)

There is an i, such that bi*r C C G H { I 2 ) (B)zi=b i

(χ)

for some Zi e FCH{D)CH(D). τ{Β) in DCG(D)/D strongly covers τ(ϊ,) in DC Η (D) / D for any i (or some i), where τ is the homomorphism induced by the canonical map

DCG(D)

DCG(D)/D.

In the case where the subgroup is normal, strong covering has a clear relation to general covering. Proposition 2.4. Let Η be a normal subgroup of G. Let Β and b be blocks of G and Η respectively. Then Β strongly covers b if and only if Β covers b and Β is weakly regular with respect to H. As a corollary, we can get a result which is a special case of Corollary 4 in [B12], Corollary 2.5. Let Η be a normal subgroup of G, and B, b blocks of G, Η respectively. Then bK%G = Β if and only if fcextG = B.

3. Brauer's Third Main Theorem under Extended Induction For ρ-regular induction, therefore Brauer induction, the following result is known as Brauer's Third Main Theorem: Theorem. Let Η < G, Β be an RG-block and b an RH-block such that bG = Β (or i>regG = Β). Then Β is the principal block of RG if and only if b is the principal block of RH. (See [B] Corollary 2). For Alperin-Burry induction, it is easy to see that the principal block always induces to the principal block if the induction is defined. But the property no longer holds for extended induction (see [W] Example 2.10). Fortunately, for

On Extended Block Induction

89

ρ-solvable groups, extended induction does have the property stated above for Alperin-Burry induction. Theorem 3.1. Let G be a p-solvable group, Κ < G. Then the principal block BQ of G strongly covers the principal block bo of K. Hence if b™tG is defined then fcgXtG = BQ. Remark. Brauer's Third Main Theorem describes the behavior of principal blocks under the Brauer correspondence of blocks. The theorem above shows that, for p-solvable groups, principal blocks induce to principal blocks in the extended sense if the induction is defined. However, even in p-solvable groups, principal blocks may be induced in the extended sense by some non-principal blocks of subgroups. Example 3.2. Let k = GF(9). Let generator v. Let

be the multiplicative group of k with

Then Η = Z 4 and |S| = 9. Let G — S · Η < SL(2, 9). Then G is a solvable group of order 36. Let ρ = 3. Then G has two 3-blocks: BQ and B\.

-1.

Let bß be the block to which β belongs. Then bß is not the principal block of Η and using Proposition 2.2 we can verify that BSßTG = BQ. This gives an example where a non-principal block induces in the extended sense to the principal block of a solvable group. Therefore extended induction is properly weaker than ρ-regular induction, even in solvable groups.

4. A Class of Infinitely Many Counterexamples to the Transitivity of Extended Induction It is easy to see by definition that the transitivity property of Brauer block induction and ρ-regular induction holds along blocks. That is, if Τ < Η < G is a chain of three groups, and b, b and Β are blocks of Τ, H, and G respectively, such that b induces to b, and b induces to Β in Brauer's sense (or ρ-regularly), then the induction of b in G is defined and equal to B. Ellers

90

G. Huang

showed that Alperin-Burry induction does not have the transitivity property (see [E]). However, general covering obviously has the transitivity property along blocks, therefore, if the induction of b in G is defined, then it must be B. Now we would like to know whether the transitivity property holds under extended block induction. Unfortunately, the answer is negative. Let ρ = 2, η > 3 be an even integer. Let G = GL(n, 2) and

Let GF(2"~') # = (χ). Then under the multiplicative action on the vector space GF(2" - 1 ), χ can be regarded as an element of Η of order 2 n _ 1 — 1. Let q be a Zsigmondy prime number with respect to (2, η — 1). Namely, q is a divisor of 2" _ 1 — 1 but not a divisor of any 2 r — 1 with r < η — 1. Let Τ = (ζ) € Syl 9 ((x)). Then we have a chain of three groups: Τ < Η A* and ψ : G B* are the homomorphisms that define the interior G-algebra structures of A and Β respectively, then a map / : A — Β is an interior G-algebra homomorphism if / is a homomorphism of algebras and if

f(o which have a defect group isomorphic to P. For b e "Bp, let \b\ = n if b is a block of Sn.

Theorem 2.3.1. 1.

There exists an integer Ν ρ with the following property: For any b in Bp with n = \b\ > Ν ρ there exists c in Bp with m = |c| < \b\ such that there is an idempotent I of (0Snb)Sm for which 0Smc ®o MatL(O) ~

WSnbI

On Blocks and Source Algebras

99

as interior Sm algebras. In particular, there is an interior Ρ-algebra isomorphism between a source algebra of b (considered as a block of Sn) and a source algebra of c (considered as a block of Sm ). 2.

The statement of (1) remains valid on replacing the group Sn by An and the group Sm by Am. As an immediate corollary of the above theorem and of Corollary 2.2.2 we

get Corollary 2.3.2. Puig's conjecture holds for the ρ-blocks of the groups Sn, for every odd prime p. Corollary 2.3.3. Puig's conjecture holdsfor the faithful ρ -blocks of the groups An, for every odd prime p. Remark 2.3.4. A result of Puig and Lincklemann on central ρ extensions proves Corollary 2.3.2 for the case ρ = 2 as a consequence of Corollary 2.2.2 Remark 2.3.5. For η > 3 and η φ 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 the 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.I., 1980, 369-375.

[Β]

M. Broue, Theorie locale des blocs, in: Proceedings of the International Congress of Mathematicians, Berkeley, 1986, 360-368.

[C]

M. Cabanes, Local structure of the ρ-blocks of Sn, Math Z. 198 (1988), 519-543.

[H]

J. F. Humphreys, Blocks of projective representations of the symmetric groups, J. London Math. Soc. (2) 33 (1986), 441-452.

[HH]

P. N. Hoffman and J. F. Humphreys, Projective Representations of the Symmetric Groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1992.

[J]

G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math. 682, Springer-Verlag, Berlin 1978.

100

Radha Kessar

[Κ]

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.

[Ml]

A. O. Morris, The spin representation of the finite group, Proc. London Math. Soc. (3) 12 (1962), 55-76.

[M2]

A. O. Morris, On ß-functions, J. London Math. Soc. 37 (1962), 445^*55.

[O]

J. Olsson, On the ρ-blocks of the symmetric and alternating groups and their covering groups, J. Algebra 128 (1989), 188-213.

[PI]

L. Puig, Pointed groups and construction of characters, Math Z. 176 (1981), 358-369.

[P2]

L. Puig, Local fusions in block source algebras, J. Algebra (1986), 358-369.

[P3]

L. Puig, On Joanna Scopes' criterion of equivalence for blocks of symmetric groups, Algebra Colloq. 1:1 (1994), 25-55.

[S]

I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitionen, J. Reine Angew. Math. 139 (1911), 155-250.

[Sc]

J. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), 4 4 1 ^ 5 5 .

Department of Mathematics Yale University New Haven CT 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 ρ 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 Broue 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, ρ is a prime number and G is a complete discrete valuation ring having an algebraically closed residue field k — 0 / 7 ( 0 ) of characteristic ρ (we allow the possibility 0 = k), and all the modules we consider are finitely

102

Lluis Puig

generated over Ö. Let G and G' be finite groups, b and b' be respectively blocks of G and G' (i.e. primitive idempotents in Ζ(OGb) and Z{ÖG'b')) and finally PY and Ργ> be respectively maximal local pointed groups on A = OGb and A' = 0G'b' (i.e. Ρ and P' are respectively defect groups of b and b', whereas γ and γ' are respectively conjugacy classes of primitive idempotents i in Ap and i' in ( A ' ) p such that i φ AQ and i' · (iAi)* mapping u e Ρ on ui (see [10], §2 for more detail on the notation). 1.3. First of all, let recall the current cases we know where b and b' are Morita equivalent; they are so in the following three situations: 1.

The blocks b and b' are nilpotent and the defect groups Ρ and P' are isomorphic ([9], Main theorem).

2.

The groups G and G' are ρ-solvable and the groups obtained by iterating Fong's reduction are isomorphic ([11]).

3.

The group G is a Chevalley group over a finite field of characteristic different from ρ and, for some parabolic subgroup Η of G, some block e of Η such that eb = e , some Levi complement L in Η of the radical of Η and some block / of L such that f e — e, formed by cuspidal irreducible characters, we have G' = NG(L, f),b' = / , Ρ' = Ρ c L ([12], Corollary 5.10).

In all these situations, it happens that Ρ and P ' are isomorphic, so that we may assume that Ρ = Ρ ' , and that there is an indecomposable OP-module Ν of vertex Ρ such that, setting S = Endo(N) and considering it as an interior Ρ-algebra, we have an interior Ρ-algebra embedding Ay-^S®0A'Y,

(1.3.4)

(i. e. injective homomorphism with image i"(S ®0 Α'γ,)ΐ" where i" is the image of the unity element) and Ρ stabilizes an O-basis of S. Actually, it is not difficult to prove that, conversely, the existence of such embedding forces the symmetric one A'y

Ay

and implies that b and b' are indeed Morita equivalent.

(1.3.5)

Morita and Rickard Equivalences

103

1.4. On the other hand, according to a well-known definition, b and b' are Morita equivalent if (and only if) there is an Ο-free 0 ( G χ G') -module M" associated with b (b')° such that, denoting by (M")* the dual Ö-module HomoiM", Ö) which is an O-free 0 ( G χ G')-module too, associated with b° φ'), we have repectively Ö(G χ G) and 0 ( G ' χ G')-module isomorphisms M" ®0G/ (M")* = A and (M")* 0G M" = A',

(1.4.1)

and our first purpose had been to connect these ismorphisms with those embeddings. A first remark in that direction is that these module isomorphisms can be easily modified to obtain algebra isomorphisms; indeed, recall that isomorphisms 1.4.1 imply that, in particular, the restriction of M " to 0(1 χ G') is projective, so that we get an G(G χ G)-module isomorphism Λ = End0(M")lxG'

(1.4.2)

but, it is easily checked that this isomorphism ought to be the structural interior G-algebra homomorphism Λ —> E n d 0 ( M " ) l x G '

(1.4.3)

multiplied by some element in E n d o ( M " ) G x G ' , which forces homomorphism 1.4.3 to be bijective. 1.5. Another easy consequence of isomorphisms, 1.4.1 is that M " is an indecomposable O(GxG')-module; hence, ithas a vertex P " and an OP"-source N", and if we are interested in relating to each other the local structures of A and A', it is reasonable to employ the local structure of M" to get it. So, set S" = Endo(W') considered as an interior P"-algebra and recall that the induced interior G χ G'-algebra I n d G x G ' ( S " ) ([10],2.14) is just Endo(Ind G ,f G (N")) in that case; now, since M " is a direct summand of the induced 0 ( G χ G')-module Ind G x G '(W"), we get from 1.4.3 an interior G-algebra embedding A — • (IndGxG'(S"))lxG'

(1.5.1)

and to "compute" the second term, we have been led to introduce the noninjective induction of interior //-algebras.

104

Lluis Puig

2. The Noninjective Induction 2.1. Let Η and H' be finite groups, φ : Η —> Η ' a group homomorphism and Β an interior //-algebra. Set Κ = Ker(ip) and consider Β as an 0(K χ AT)-module by left and right multiplication; the tensor product 0 ® o κ Β has an evident right Β -module structure Ο ®οκ

Β χ Β —•

Ο ®οκ

(2.1.1)

Β

which is clearly compatible with the action of Κ by conjugation on Β and on Ο ®oκ Β; moreover, if a, a' € Β and Κ fixes 1 a in Ο ®QK Β then, for any χ e Κ we get (\®a)-(x-a') = l®aa'

(2.1.2)

consequently, the map 2.1.1 restricted to { Q ® Q K B ) inducesanew bilinear map k

(Ο ®οκ Β)κ χ (0 ®οκ B) —> G ®οκ Β

-stable (2.1.3)

which determines a product in (0 ®οκ B)K (Ο ®οκ Β)κ χ (0 ®οκ B)K —> (0 ®0K B)K

(2.1.4)

mapping (1 ® a, 1 ® a') on 1 ® aa! for any a, a' e Β such that Κ fixes 1 ® a and 1 ® a' in 0 ®0κ Β. It is easily checked that (0 Oqa: B)K with this product becomes an associative 0-algebra and that the structural map Η —> Β* induces a group homomorphism Η = Η/Κ

(Ό®οκ B)K.

(2.1.5)

2.2. We are ready to define the induced interior //'-algebra Ind^,(5); considering (0 ®οκ B)K and 0 H ' as Ö(Η χ H)-modules, the second via φ, we set I n d ^ ß ) = OH' ®Όΰ (0 ®οκ B)K ®oü

QH'

(2.2.1)

with the distributive product defined by χ' ® (1 ® a · ζ • a') ® t' (x' ®(l®a)®

s')(y' ® (1 ® a') ®t')=

0r

(2.2.2)

0 according to whether or not there is ζ € Η such that φ(ζ) = s'y', for any χ', s', y', t' G H' and any a, a' € Β such that Κ fixes 1 ® a and 1 ® a' in 0 ®οκ Β, and with the group homomorphism H' —OH'

®0fj (0 ®οκ B)K ®0jj OH'

(2.2.3)

Morita and Rickard Equivalences

105

mapping x' e H' on £ , x'y' ® (1 ® l ß ) ® y' where y' runs over a set of representatives for Η /φ(Η) in Η'. Now it is easy to check that I n d p ( ß ) is indeed an interior H'-algebra. When Κ — 1 it reduces to the ordinary induction, noted Ind^ (Β) ([10],2.14), and when φ is surjective we identify it with (Ο ® ο κ B ) K . 2.3. Actually, as in ordinary induction, the induced interior //'-algebra mimics the structure of the full 0-linear endomorphism algebra of an O H ' -module induced from an O-free OH-module. More generally, if Ν is a Β -module then Ιηάφ(Ν) = ΌΗ' ®0H Ν becomes an Ind^, (5)-module with the distributive action defined by (χ' Θ (1 ® α) ® s') • ( / ® n) =

χ' ® a · (z • ή) or

(2.3.1)

0 according to whether or not there is ζ € Η such that H ' a second group homomorphism, and consider the following pull-back diagram of groups

Η

L

Ψ — » •

Η'

Ψ —> L'

It is clear that, up to suitable identification, φ and ψ have the same kernel K ·, from this fact it is quite clear that we have a canonical interior L ' - a l g e b r a embedding

l n d f (Res„(*))

— • Res^(Ind^(5))

(2.6.2)

which is an isomorphism if and only if Η ' = φ ( Η ) · η ' ( Ζ / ) . More generally, χ if we consider all the pull-backs determined by the family {(η') }x', where x' runs on a set of representatives for η ' ( ϋ ) \ Η ' / φ ( Η ) in Η ' , then the very Mackey formula states that the image of the unity elements of the corresponding interior L ' -algebra embeddings form a pairwise orthogonal idempotent decomposition of the unity element in Ind^(ß).

Morita and Rickard Equivalences

107

2 . 7 . There is no difficulty on inducing any homomorphism between two interior if-algebras to obtain an interior Η'-algebra homomorphism between the corresponding induced interior if'-algebras, and on proving that Indy, becomes a functor between the categories of interior H - and //'-algebras. Then, all the homomorphisms above are in fact natural maps between suitable functors and, to be complete, it has to be checked all the obvious compatibilities between them, which amounts to prove the commutavity of certain diagrams. 2 . 8 . Finally, let us consider the relationship between the Brauer section B(Q) of Β at any ρ-subgroup Q of Η (i.e. the interior C#(Q)-algebra k ® ο (B® / R B R ) where R runs on the set of proper subgroups of Q) and the corresponding Brauer section ( I n d ψ ( Β ) ) { φ ( ζ ) ) ) of Ind^(ß). It seems hopeless to obtain a precise general answer but if we assume that φ is surjective and that, considering the action of Η χ Η on Β by left and right muliplication and denoting by Δ(Η)) the diagonal subgroup of Η χ Η, Β is a direct summand of a permutation Ö((ÄT χ Κ ) · Δ (//))-module with projective restriction to Ö(K χ Κ) we get the following result: 2.8.1. If Q' is a p-subgroup of H' then the natural homomorphism BK(Q') —• (Ο ®οκ B)K(Q') determines, for any complement Q of Κ χ in φ~ (ζ>') an interior B, it is quite clear that g(Do) = g(D) and it can be proved that for any O-algebra homomorphism f : D ®o D —> B, the composed homomorphism f

Δη

Do - A Do 0 0 Do C D Β (4.3.5) 4.4. Consequently, if Β and B' are interior DH-algebras and we denote by stß : DH —> Β and stß' : DH —>· B' the structural maps, the tensor product of Β and B' is the O-algebra Β ο B' endowed with the O-algebra homomorphism DH —> Β ®o B'

(4.4.1)

mapping χ e Η on st 5 (jc) ® st ß /(x) and g e D on A*B9B,(stB ® stB')(g). When Β — Endo (Μ) and Β' = Endo (Μ'), we get the tensor product of DH -modules; more generally, it is easily checked that if L is a normal

Morita and Rickard Equivalences

113

subgroup of Η then the kernel of the canonical map Μ ® o Μ ' —> Μ ® 0 L M '

(4.4.2)

where the right OL-module structure of Μ is given by y * m = j - 1 · m for any y e L and any m e M, is a DH-submodule of Μ o M' and therefore Μ M' becomes a D(H/L)-module. The associativity of the tensor product comes from the coassociativity of Δο and the uniqueness of the extension 4.3.5; similarly, the commutativity "up to isomorphism" comes from the equality ί 0 ο Δ 0 = int(fo) ο Δο

(4.4.3)

where ίο is the automorphism of Do o A) which exchanges both factors, to € Fo ® o F0 maps (z, z') on (—l) zz and int(io) denotes the corresponding inner automoφhism of Do ® o A)· Moreover we consider the Ö-algebra isomorphism t : defined by t(d) — sd and ( t ( f ) ) ( z ) = f(—z) which fulfills over DQ

(4.4.4) for any ζ € Ζ and any / e F

(t ® t) O so Ο Δο = Δο O t

(4.4.5)

and allow us to define the opposite interior DH-algebra B° of an interior D//-algebra Β and, consequently, the O-dual M* of an O-free DH-module Μ . In particular, notice that an interior DH-algebra Β has a canonical D(H χ H)-module structure obtained from the Ö-algebra homomorphism Β ® o B° —> Endo (Β)

(4.4.6)

defined by left and right multiplication. 4.5. Let Β be an interior DΗ-algebra; as when working with DH-modules, we have to consider the so-called cycles and bords of B\ actually, for our purpose here, we can restrict ourself to consider only 0-cycles and 0-bords. It is not difficult to check from the D-module structure of Β that the set Co(B) of 0-cycles of Β is the interior 0//-subalgebra formed by the elements of Β which centralize the image of D by the structural map st# : DH —> Β and the set BQ(B) of 0-bords of Β is the ideal of CQ{B) formed by the elements d -a + a-d where a runs on the set of elements of degree one of Β; as usual, we set H0(B)

which is an interior OH -algebra.

= Cq(B)/BO(B)

(4.5.1)

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Lluis Puig

4.6. More generally, if L is a subgroup of Η then BH is an interior D C / / ( L ) algebra and we consider also C0(BL)

= C0(B)L,

B0(BL)

C B0(B)L

and

H0(BL)

=

CQ{BL)/BQ(BL).

(4.6.1) Now a point e of L on Β is a conjugacy class of primitive idempotents in CQ{BL) = CQ(B)L, so it is just an ordinary point of L on the interior OH-algebra CQ(B) ([10], 2.10) and we say that e is contractile if e c B0(B)·, notice that if j e e then jBj endowed with the map DL —• j Bj defined by the multiplication by j is an interior DL-algebra, noted BE and obviously we have ([10], 2.13) CO(BE)

= C0(B)E.

(4.6.2)

As in the interior OH-algebra case, if Β = Endo (Μ) then BE = Endo Ο'(Μ)) and j(M) is an indecomposable direct summand of the DL -module Res^(M). Coherently, pointed groups on Β are nothing but ordinary pointed groups on CQ(B) and we define inclusion and localness from those on CQ(B) ([10], 2.10); except that there are the contractile pointed groups which we are not interested in. Notice that a pointed group on Β contained in a contractile one is itself contractile and it can be proved that a pointed group on Β is contractile if and only if its defect pointed groups are so. 4.7. However, localness on interior DH-algebras is not so plain; indeed, if β is a ρ-subgroup of H , it is clear that the Brauer section B(Q) of Β at Q inherits an interior D C / / ( ö ) - a l g e b r a structure from B, so that we can consider the interior ÖC//(Q)-algebra CQ{B{Q)). On the other hand, according to our definition, the set of local points of Q on Β corresponds bijectively with the set of points of the Brauer section (CQ{B)){Q) of CQ{B) at Q ([10], 2.10.1). It is easily checked that we have a canonical interior CH(Q)-algebra homomorphism ( C 0 ( ß ) ) ( ß ) — • CO(B(Q))

(4.7.1)

but it needs not be either injective or surjective; in particular, a local point of Q on Β needs not to determine a point of 1 on B{Q). 4.8. There is a particular situation, which occurs in Rickard equivalence between blocks (see Section 5 below), where homomorphism 4.7.1 is an isomorphism. First of all, recall that a DH -module Μ is contractile if an only if id Μ belongs to 5 o ( E n d o ( M ) ) or equivalently, all the points of Η on E n d o ( M ) are contractile. On the other hand, any interior if-algebra can be considered

Morita and Rickard Equivalences

115

as an interior D Η-algebra via the canonical Ö-algebra homomorphism μ : D —• Ο

(4.8.1)

mapping / e F on / ( 0 ) and d on zero; in particular, any Ö//-module can be considered as a DH-module and we call it a «-restricted DH-module. Now, we call the direct sums of contractile and «-restricted DH-modules 0-split D/f-modules and it is not difficult to prove that if Β is a 0-split D ß - m o d u l e then we have (Co(Ä))(ß) = Co(fl(ß)) and for any subgroup L

B{Q) ,

L

of

containing

NN(Q)

Q

(4.8.2) such that

BR^B1)

=

we have also Br*(C0(BL))

= CQ(B(Q)L).

(4.8.3)

4.9. Finally, it remains to introduce the induction of interior DH-algebras. Let H' be a second finite group, φ : Η —> Η' a group homomorphism and Β an interior DH -algebra. Set Κ = Ker( Β the structural map; according to Section 2, we have already the interior OH-algebra ( 0 ®οκ B)K ®0„

Ind^(ß) = 0 Η '

OH',

(4.9.1)

and we extend to D H ' the structural 0-algebra homomorphism O H ' — • Ind^(ß) by mapping g e D on Tr^('W)(l(g>(l(g)stß(g)) Indf (Be)

(4.9.5)

such that the corresponding extended diagram is commutative too.

5. Rickard Equivalences between Brauer Blocks 5.1. Let us come back to our standard notation in 1.2 and consider respectively A and A' as interior OG- and OG'-algebras, via the structural maps induced by homomorphism 4.8.1; in particular, they become respectively D(G χ G)and D(G' χ G')-modules 4.4.6. In [16], Jeremy Rickard proves that the socalled derived categories of the categories of A- and A'-modules are equivalent, as triangulated categories, to each other if and only if: 5.1.1. There is an indecomposable D(G χ G')-module M" associated with b Ρ' the respective retrictions of π and π', a local point γ of Ρ on Ind ff (S" ®o ResCT'(A'y/)) such that IndCT(S" o is a 0-split D(P χ P)-module and we have an interior Ρ -algebra isomorphism AY = H0(lnda(S"

o R e s , , / ^ ; , ) ) ^ ·

(5.5.1)

Remark 5.6. Actually, as in Remark 3.2, there is a more precise result involving multiplicity modules which gives a necessary and sufficient condition on M" for inducing a Rickard equivalence between b and b'. 5.7. From now on, we assume that M" induces a Rickard equivalence between b and b'\ then, in 5.4 we may choose Py in such a way that: 5.7.1. We have π{Ρ") embedding hYf'y

= Ρ, π'(Ρ")

= Ρ' and an interior

DP-algebra

: Ä9 —• Inda (S" ® 0 Re^(A' y ,))

where γ is the noncontractile point of Ρ on λ such that Ay = ffo(Äp) (cf. 5.3.2). In order to give a more precise description of the relationship between the local categories of b and b' (cf. 3.4), it is proving useful to consider the following definition (which has not been introduced in Section 3 to avoid too much redundancy). A triple (Q$, Q's,) of local pointed groups Qs on A, on A" and Q'&l on A' is called a local tracing triple on A, A" and A' if we have π(ζ>") = Q, n'(Q") = Q' and a canonical interior D Q-algebra exoembedding (cf. Remark 5.2 below) hf·*' : A-& — • I n d < 8 >

0

Resr/(^,))

(5.7.2)

Morita and Rickard Equivalences

119

where r : Q" — • β and x': Q" —> Q' are respectively the restrictions of π and π', and δ isthenoncontractilepointof β on Λ suchthat As = Ho(A^) (cf. 5.3.2). Notice that ( Ρ γ , Ργ„, Ρ',) is indeed a local tracing triple on A, A" and A', since A^„ = S". s o o n Cq(A"),

Recall that 8 is also a point of β χ G' on

and let us denote respectively by Vcq(A")-&(Q's»)

anc

*

A",

Va'(Q's>)

2

simple modules over C 0 ( A " ) p " and ( A ' ) ' where 8" and 8' act nontrivially. 0

Theorem 5.8. A

triple

Qg„ on A" and and only if Qy, n'(Q") from of

( Qs, Q's'„, Q's,) of local pointed

Q'&, on A' is a local tracing is a defect pointed

= Q' and, considering F

OCG>(Q )

Qs

on

to ( A ' ) ' and to Cq(A")9",

the structural

maps

g' and

R e s ^ V ^ ' ^ / ) ) is a

A,

A' if

group of (Q χ G')^ on A", we

respectively

0

groups

triple on A, A" and

have g"

quotient

ReSgKVco^iß^)).

Remark 5.9. Notice that neither exoembedding 5.7.2 nor Theorem 5.8 are symmetric in the roles of A and A'; indeed, if (Qs, Qg„, Q's,) is a local tracing triple on A, A" and A' then ( Q'&,, Q'^,, Q$) needs not to be a local tracing triple on A', (A")° and A (i.e. with repect to the Rickard equivalence between b' and b induced by (M")*). However, if (Q' &/ , R"„, Re ) is a local tracing triple on A', (A")° and A, it can be proved that there is χ e G such that (Qs)x C Re. 5.10. It is clear that G χ G' acts by conjugation on the set of local tracing triples on A, A" and A'. Moreover, if (Qs, Qg„, Q'&,) and (Re, R"„, R'e,) are local tracing triples on A, A" and A', we say that (Rf, R"„, R'f,) is contained in (Qs, Q's//, β ^ , ) if we have Re c Qs on A, C Q's'„ on A", R'e, c Q'&, on A' and, for a suitable canonical exoembedding (cf. Remark 5.12 below) of interior D - a l g e b r a s , the following diagram is commutative Resg(Af·4') Res^(A^)

J

>

-«",«'

ε

A^

Res 2 (Ind T (A^, ® o Res T '(Ay)))

8 ε",ε'

•

(5.10.1)

Ind v (Ag„ ® o Res w /(A^))

where ν : R" —> R and v' : R" —> R' are respectively the restrictions of 7Γ and π'. Then it is easily proved that the inclusion between local tracing

120

Lluis Puig

triples is transitive and we consider the category where the objects are the local tracing triples on A, A" and A', and the morphisms are the group exomorphisms between the middle terms induced by composing the suitable restrictions of inner automorphisms of G χ G' and the inclusions of local tracing triples: we call it the local category of the Rickard equivalence induced by M". Theorem 5.11. The functor from the local category of the Rickard equivalence induced by M" to the local category of the block b defined by the first projection is full and induces a bijection between the sets of isomorphism classes of objects in both categories. Remark 5.12. Actually, to give a precise definition and to prove the uniquest

sff

p/

ness of the canonical exoembeddings h- ' in 5.7.2 and g,„', in 5.10.1, we 8 c ,6 need to introduce the Higman envelope of an interior DH-algebra B, namely the smallest interior DH-algebra £ ( 5 ) admitting interior DH -algebra exoembeddings e(B) : Β —E(B)

and hv :

—E(B)

(5.12.1)

for any local pointed group Τφ on B, such that the following evident diagram of exoembeddings is commutative ([10], 2.13.1 and 2.14.1) Res^ (Β)

Res^ (e(B)) •

Res^ (E(B)) Res «(h9)

h

-Brauer pairs then a group exomorphism from

122

Lluis Puig

R to Q belongs to the Brauer category if and only if it is induced by χ e G such that ( R e ) x c Qs for suitable local points e of R and δ of Q on A such that b(e) = g and b(5) = / . 6.4. Let Qs be a local pointed group on A, ( Q s , Q$„, Q'&,) a local tracing triple on A, A" and A' and R a normal subgroup of Q, and denote respectively by r : Q" —• Q and τ' : Q" —> Q' the restrictions of σ and σ'. It is not difficult to see that our basic hypothesis implies that Q stabilizes an 0 -basis of by conjugation and, consequently, that equality 4.8.3 applies; hence the unity element is primitive in Cq(A^(R))^ or, equivalently, BR£(8) is contained in a point of Q on A ( R ) , which actually is a local point. On the other hand, always from our basic hypothesis, statement 2.8.1 suitably modified replacing Ö by D applies to the interior D Q"-algebra A^,®oRes r '(A^) and to the surjective group homomorphism r : Q" —> Q. It follows quite easily that: 6.4.1. the

There

is a section

μ

interior D C q ( R ) - a l g e b r a h f '

' ( R )

S

:Ä

r-1(/?)

—>

: R

r e s t r i c t i o n of

τ

such

that

exoembedding

— • (IndT(A£„ ® 0 Res r >(Aj,)))(Ä)

^ R )

0 ResT/(A^,))

factorizes via

of the

in

an

i n t e r i o r D C q ( R ) - a l g e b r a ex-

oembedding

h? : A ^ R ) — • Ι η ά τ μ ( Α ^ μ ) ®k R e s r / ( A j , ^ ' ) ) ) where μ' = τ' ο μ, μ ' ,

and

τ

μ

: C

Q

Rß

and Rß

" ( R ^ )

C

( R )

Q

are respectively the images of μ and τ'μ

and

: C

Q

» { R — >

C q ' ( R ^ ' ) are

respectively the restrictions of r and τ'. Notice that μ' is injective since {0} φ A y , ( R ) = η Ker(r')))(fl M ) implies R » η Ker(r') = (1, 1). ß

6.5. Now, as in 5.7, choose Py, in such a way that ( Ρ γ , Ρ' γ ,) is a local tracing triple on A, A" and A'; applying statement 6.4.1 to this triple, with R = P, we get once for ever a section λ : Ρ" —> Ρ of σ and then it can be proved with the notation of 6.4 that: 6.5.1

For

any

(β")(*.*') c U € R ,

χ P " ,

e

G

such

that

( Q \ b(S')) ' x

( Q s )

C

x

C

P

there

y

( P ' , b ( y ' ) ) and

X ( u

is x

) =

x'

€

G'

such

μ(«)(*·*'>f o r

that any

but we know nothing about ( S ' ) · · * ' ) and ( S ) ' . In particular, for any local pointed group Qs on A such that Qs C PY, we can choose a local tracing ( x

r

x

Morita and Rickard Equivalences

123

triple ( ß a , ߣ„, Q's,) on A, A" and A' such that Q" c P", (Q',b(S')) c (P', b(y')) and the restriction of λ to β fulfills condition 6.4.1 for R = β ; then applying again statement 6.5.1 to those choices, it is not difficult to prove the following result. We set λ' = σ' ο λ and, for any έ>-Brauer pair ( β , / ) such that ( β , / ) C (Ρ, b(y)), we denote by β λ ' the image of β by λ' in P' and by / λ ' the block of CG'(QX') suchthat (QX\fK')C(P',b(Y')).

(6.5.2)

Theorem 6.6. With the hypothesis and the notation above, the group homomorphism λ' : Ρ —> Ρ' is bijective and induces an equivalence between the Brauer categories of b and b' mapping any b-Brauerpair (β, / ) contained in (P,b(Y)) on ( β λ ' , / λ ' ) · 6.7. Finally, we claim that in Theorem 6.6, the blocks of / of C c ( ß ) and / λ of CG'(Qa ) over k are basically Rickard equivalent too. Indeed, for any fr-Brauer pair, we can choose a G-conjugate ( β , / ) such that ( β , / ) C (P, b{y)) and that Cp(Q) is a defect group of / ; then it is not difficult to prove from Theorem 6.6 that Cpi{Qx ) is a defect group of / λ ; in particular, if € and € λ are respectively the local points of R = Q • Cp(Q) on A and of Ry = QX' • C P / ( ß x ' ) on A' such that Re C Py and R^, C P', then AE(Q) and Α^ λ ,(β λ ) are respectively source algebras of / and / λ ' . On the other hand, choosing a local tracing triple ( R e , R"„, R'e,) on A, A" and A' and applying statement 6.4.1 to the normal subgroup β of R, statement 6.5.1 allow us to modify eventually our choices in such a way that we have X(R) c R" C P" and (Rk',b(€X'))

C (R', b(e')) C (Ρ',Κγ'))

(6.7.1)

and an interior DCp{ Q)-algebra exoembedding Α | ( β ) —> Μ τ ( Α ^ , ( β λ ) ® k R e s r ' ( K , ( Q x ' ) ) )

(6.7.2)

where β λ and β λ ' are respectively the images of β by λ and λ', and r : C/j//(ß x ) — • CP(Q) and τ' : CR»{QX) —• CR>{QX') the restrictions of σ and σ'. 6.8. Actually, since the blocks b(ex ) and b{t') are nilpotent ([2], § 1), the inclusion of ö-Brauer pairs in 6.7.1 imply C R'€, C /»;, and CRAQX')

= Q ' ( ß V ) = C^(ß X ')(6.8.1)

Moreover, although R"„ needs not be contained in Py,, there is (x, x') in G χ G' such that (R"„)(x'x''1 C P"„ and therefore, it follows from our basic

124

Lluis Puig

hypothesis that Α"„(β λ ) is a simple fc-algebra, so that Α"„(β λ ) = Endk(M'f'„)

(6.8.2)

where M"„ is a simple Α"„(β λ )-module, in particular, M"„ becomes a DCR„(Qk)- module and although it needs not be indecomposable of vertex CRn{Qx ) it can be proved, from exoembedding 6.7.2 and Theorem 5.5, the following result Theorem 6.9. With the hypothesis and the notation above, there is a subgroup T" of CR"(Qx) and an indecomposable direct summand NQ of Res^f x

(βλ)

Cp'(Q )

(Λ^'„) of vertex T" such that τ {Τ") = CP(Q), and we have an interior DC ρ (Q)-algebra

=

exoembedding.

At(Q) — • Ind P (S% ®k R e s p ' ( A ^ ( Q k ' ) ) ) where SQ = End*(W£) and Ρ : Τ" —• CP(Q)

x'{T")

(6.9.1)

and Ρ' : T" —• C P / ( ß x ' )

are respectively the restrictions of τ and τ'. In particular, a suitable indecomposable direct summand of

\N'Q) induces a basic Rickard

equivalence between the blocks f of kCa(Q)

and /λ'

of

kCc'iQ^')·

Remark 6.10. If NQ lifts to an O-free endopermutation ΟΓ"-module NQ (i.e. suchthat Τ" stabilizes an 0-basis of Endo (Ng) too) thenRickard's construction in [18], §5 can be easily generalized to prove that an indecomposable direct summand of Indr,

°

(N Q k )

induces a basic Rickard equivalence between / and / λ ' . Remark 6.11. We know that

= R' whenever R maps surjectively onto

OP(NG(Re)/R-CG(R))· otherwise we know nothing about the inclusion Rx' c R'·

Morita and Rickard Equivalences

125

References [1 ]

Laurence Barker, G-algebras, Clifford Theory and the Green Correspondence, J. Algebra 172 (1995), 335-353.

[2]

Michel Β roue, Les i -blocs des groupes GL(n,q) et U(n,q2) et leurs structures locales, Asterisque 133/134 (1986), 159-188. Michel Broue, Isometries parfaites, types de blocs, categories derivees, Asterisque 181/182 (1990), 61-92. Michel Broue, Equivalences of blocks of group algebras, Finite dimensional algebras and related topics (V. Dlab et al., eds.), Proc. Internat. Conference Representations of Algebras and Related Topics, Ottawa, Canada, 1992, Kluwer, 1994, 1-26. Michel Broue, Rickard Equivalences and Block Theory, Groups '93 GalwaySaint Andrews, London Math. Soc. Lecture Notes 211, Cambridge Univ. Press, 1995 ,58-79. Markus Linckelmann, Derived equivalence for cyclic blocks over a ρ -adic ring, Math. Z. 207 (1991), 293-304.

[3] [4]

[5]

[6] [7]

Markus Linkelmann, The isomorphism problem for blocks with cyclic defect groups, Invent. Math. 125 (1996), 265-283.

[8]

Lluis Puig, Local Fusions in Block Source Algebras, J. Algebra 104 (1986), 358-369. Lluis Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77-116. Lluis Puig, Pointed Groups and Construction of Modules, J. Algebra 116 (1988), 7-129. Lluis Puig, Block source algebras in ρ-solvable groups, Letter to Morton Harris, 1993. Lluis Puig, On Joanna Scopes' Criterion of Equivalences for Blocks of Symmetric Groups, Algebra Colloq.l (1994), 25-55. Lluis Puig, On Thevenaz parameterization of interior G-algebras, Math. Z. 215 (1994), 325-355. Jeremy Rickard, Morita Theory for derived categories, J. London Math. Soc. 39 (1991), 37-48. Jeremy Rickard, Derived categories and stable equivalences, J. Pure Appl. Algebra 61 (1989), 307-317. Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37-48. Jeremy Rickard, Finite group actions and etale cohomology, Publ. Math. IHES 80(1994), 81-94. Jeremy Rickard, Splendid equivalences: derived categories and permutation modules, preprint 1994.

[9] [ 10] [11] [12] [13] [14] [15] [16] [17] [18]

126

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[ 19]

Klaus Roggenkamp, Subgroup rigidity of p-adic group rings (Weiss arguments revisited), J. London Math. Soc. 46 (1992), 432-448.

[20]

Klaus Roggenkamp and Leonard Scott, Verbal communication, Luminy, 1988.

[21]

Raphael Rouquier, From stable equivalences to Rickard equivalences for blocks with cyclic defect, Groups '93 Gal way-Saint Andrews, London Math. Soc. Lecture Notes 211, Cambridge Univ. Press, 1995.

[22]

Leonard Scott, Unpublished notes.

[23]

Alfred Weiss, Rigidity of p-adic 332.

p-torsion, Ann. of Math. 127 (1988), 3 1 7 -

CNRS, Institut de Mathematiques de Jussieu 6 Avenue Bizet 94340 Joinville Le Pont France

Some Open Conjectures on Representation Theory G. R. Robinson

Introduction Several of Brauer's questions on block-theoretic invariants (see, for example, [3]) have in recent years been placed in a more conceptual setting. Some still appear unassailable, and some have been modified or extended by later authors. We discuss here some recent developments on some of these questions.

Brauer's Problem 21 Is there a function / : Ν —• Ν such that whenever Β is a block with defect group D, and Β contains k ordinary irreducible characters, then we have | ö | < / ( * ) ? See [3] or [6]. This is a precise way of expressing the fact that a block with a large defect group should contain a large number of ordinary irreducible characters (if, as expected, the answer is positive). Little progress has been made on the most general case of this problem, though B. Külshammer recently proved that the answer is positive if we restrict attention to blocks of ρ-solvable groups. Very recently, B. Külshammer and I have used E. Zelmanov's solution of the restricted Burnside Problem to prove: Theorem ([13]). If the ρ-block Β has defect group D and satisfies the equality predicted by the Alperin-Mckay conjecture (W\), then \D\ is bounded in terms of k(B). In other words, the Alperin-McKay conjecture imples that Brauer's problem 21 has a positive answer.

128

G. R. Robinson

The " k ( B ) at Most pd" Conjecture This is Brauer's problem to prove that the number of ordinary irreducible characters in a block Β with defect group D is at most |D|. It seems fair to say that no general progress has been made on achieving this precise bound. Brauer and Feit proved that k(B) < \\D\2 + 1 ([4]), and in general the quadratic nature of the bound has not been improved. Even for blocks of ρ-solvable groups, the question appears to be very difficult. In that case, the problem reduces to the k(GV)-problem: if a p'-group G acts faithfully on a GF(/?)G-module V, does the semi-direct product GV have at most | V| conjugacy classes? Even the p - solvable case of the problem has interesting consequences for blocks of arbitrary groups. For example, Kiilshammer showed in [12] that a positive answer to the Alperin-McKay conjecture, together with a positive answer to this question for blocks of ρ-solvable groups implies a positive answer to the following conjecture of Olsson: Conjecture (J. B . 0 1 s s o n [ 16]). Let Β be a block with defect group D. Then k0(B) < [D : D'l R. Knörr ([8], [9]) did fundamental work on the k(GV) problem. Building on that, and influenced by recent work of R. Gow [7], J. G. Thompson and I proved: Theorem ([20]). Let the finite p'-group G act faithfully and irreducibly on the elementary Abelian ρ-group V. If there is some ν in V such that R e s g c ( V ) ( V ) has a faithful self-dual submodule (as GF(p)CQ(U) -module), then Gk(GV) < |V|. In my talk at the conference, I expressed the belief that such a ν would always exist, thus confirming that the k(GV) -problem would always have a positive answer. Since the conference, there have been further developments. J. G. Thompson furnished in [21] an example of a pair (G, V) for which there is no such vector υ (the prime ρ is 7). On the positive side, in [20] it is proved that if ρ is outside a certain finite set of primes, then such a vector ν always does exist. This result confirms the ρ-solvable case of the problem of this section for all but finitely many choices of the prime p. For this work and some extensions, see [19] and [20] Recently, B. Kiilshammer and I have been considering the problem of this section for principal blocks (presumably the most interesting case). We

Some Open Conjectures on Representation Theory

129

noted that an easy variant of Nagao's argument in [15] proves that whenever Ν is a normal subgroup of a finite group G, we have k(Bo P \G)) < k(B(0p)(N))k(G/N), where B(0p) denotes the principal p-block. In [11], Koväcs and I proved that there is a constant c such that when every Β is a ρ-block of defect d of a ρ-solvable group, then k(B) < cd~lpd. In trying to prove an anologous result for principal blocks of arbitrary groups, Kiilshammer and I have proved that if suffices to consider two configurations for a purported minimal counterexample G: (i)

Op/(G) = 1, Ε = E(G) φ 1, G/E is a solvable p'-group. For Ρ e Syl p (G) we have that G/ECa(P) is Abelian. Each component of Ε is normal in G.

(ii)

Op{F*{G)) = 1, and G has a unique minimal normal subgroup V such that, letting X be the full pre-image in G of Op>(G/V), G/X has the structure of case i). Either V < O(G), or V = F(G). If V < O(G), then V = X. If V £ then we have d > j [l°gp (|V„|) + l o g p (|Z(V n )|)], and CD(V0) c V0-

References [1]

J. L. Alperin, The main problem of block theory, Proc. Conf. Finite Groups, Academic Press, New York, 1976.

[2]

J. L. Alperin, Weight for finite groups, in: Proc. Symp. Pure Math 47, Amer. Math. Soc., Providence, 1987, 369-379.

[3]

R. Brauer, Representations of Finite Groups, in Lectures in Mathematics, Vol. 1, Wiley, New York (1963), 133-175.

[4]

R. Brauer, W. Feit, On the number of irreducible characters of finite groups in a given block, Proc. Math. Acad. Sei. USA 45 (1959), 361-365.

[5]

E.C. Dade, Counting Characters in Blocks, II, J. Reine Angew. Math. 448 (1994), 97-190.

[6]

W. Feit, The representation theory of finite groups, North Holland, Amsterdam, 1982.

[7]

R. Gow, On the number of characters in a block and the k(GV)-problem self-dual V, J. London Math. Soc. (2) 48 (1993), 4 4 1 ^ 5 1 .

[8]

R. Knörr, On the number of characters in a ρ-block of a /»-solvable group, Illinois J. Math. 28(1984), 181-210.

[9]

R. Knörr, A remark on Brauer's Jc(ß)-conjecture, J. Algebra 131 (1990), 444-450.

for

[10] R. Knörr, G. R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39, (1989), 48-60. [11] L. G. Koväcs and G. R. Robinson, On the number of conjugacy classes of a finite group, J. Algebra 160 (1993), 4 4 1 ^ 6 0 .

Some Open Conjectures on Representation Theory

131

[12] Β. Külshammer, A remark on conjectures in modular representation theory, Arch. Math. 49 (1987), 366-399. [13] B. Külshammer, G. R. Robinson, Characters of Relatively Projective Modules II, J. London Math. Soc. (2) 36 (1987), 59-67. [14] B. Külshammer and G. R. Robinson, Alperin McKay implies Brauer's Problem 21, to appear in J. Algebra. [15] H. Nagao, On a conjecture of Brauer for p-solvable groups, J. Math. Osaka City University 13 (1962), 35-38. [16] J. B. Olsson, Block invariants in An, An and Sn, in: Proc. Symp. Pure Math. 47, Amer. Math. Soc., Providence, 1987, 4 7 1 ^ 7 4 . [17] G. R. Robinson, Some remarks on the £(GV)-problem, J. Algebra 172 (1995), 159-166. [18] G. R. Robinson, Local structure, vertices, and Alperin's Conjecture, Proc. London Math. Soc. (3) 72 (1996), 312-330. [19] G. R. Robinson, Further reductions for the ^(GV)-problem, J. Algebra, to appear. [20] G. R. Robinson and J. G. Thompson, On Brauer's k {B) -problem, J. Algebra 184 (1996), 1143-1160. [21] J. G. Thompson, private communication. Department of Mathematics University of Leicester Leicester LE1 7RH England Email: [email protected]

Are All Groups Finite? Leonard L. Scott*

This paper is dedicated to Walter Feit on the occasion of his 65th birthday. Its contents were presented in part at the 1995 Ohio State finite group representation conference organized in celebration of that birthday. Primarily, the paper is a discussion of some classical and recent developments in the modular representation theory of finite groups of Lie type, and the problems which drive that theory. But there is also a philosophical thread . . . An old question which arose again at the conference is the following: Are all groups finite? That is, applications and broader issues aside, if we think only of our interest in finite group theory itself, is it possible to safely ignore other groups? My viewpoint is that the answer to this question has two parts: First, in representation theory, at least, we cannot ignore the infinite complex Lie groups and their characteristic ρ analogs, the algebraic groups over ¥ p . The second part of my answer is that we can, nevertheless, hope to find understandings within finite group theory and finite dimensional algebra of ideas naturally suggested by these continuous contexts, and take them further. Let me begin by convincing you of the first part of my answer: Suppose one is considering a finite group G(Fq) of Lie type, such as the special linear group SL(n,q) of degree η with coefficients in the field F 9 of q elements, q a power of a prime p. The Classification of finite simple groups asserts that almost all of the latter are variations on the finite groups of Lie type together with the alternating groups. Much earlier (1963), Steinberg [34], [35] proved all irreducible representations of G(F 9 ) with coefficients in a finite field of characteristic p, and, thus, in the algebraic closure ¥ q , come by restriction from the irreducible representations over F^ of the algebraic group G(F 9 ). The latter group is, of course, quite infinite. It is the analog via the Zariski topology, of the complex analytic Lie group G(C). Moreover, the representations we need are continuous, and even "analytic", in the sense that they are locally defined by polynomial functions. Now, the theory of finite-dimensional *

T h e author thanks N S F for its support.

Offprint from\ Representation Theory of Finite Groups, Ed.: R. Solomon © by Walter de Gruyter & Co., Berlin • New York 1997

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irreducible continuous representations of G(C) has an elegant and powerful formulation, first, in that the irreducible representations are parametrized very completely by the "theory of the highest weight" of Cartan, and, second, that the characters of these representations are known, given by the famous "Weyl character formula". See, for instance, [20] for these theories for complex semisimple Lie algebras. If we had such a parameterization and such a character formula for the finite groups G(F 9 ), not only would we would know their irreducible characters in the describing characteristic ρ of G{¥q), but we might learn something about the nondescribing case as well, where many analogies with the describing case have been discovered by Dipper and James [15], [16], [14], [21]. Working with the general linear group GL(n, q) they have found families of finitedimensional algebras, the q -Schur algebras, parameterized by a variable q, that control the nondescribing characteristic representation theory of the group GL(n, q) when q is taken to be a prime power (which may also be viewed as a kind of root of unity when the underlying field has positive characteristic), and which control the modular theory in the describing characteristic ρ when q = 1. If we knew both describing and nondescribing modular theory for the simple or nearly simple groups of Lie type, we could also hope to learn much about the maximal subgroup structure of all other finite groups [30], [2]. Indeed, this is a main organizational theme of the upcoming 1997 Newton Institute program at Cambridge. So, to summarize, it would be highly desirable to have for finite groups G(¥q) of Lie type a parameterization and character formula, as exist for the complex Lie groups G(C). Also, thanks to the work of Steinberg mentioned above, both issues for G ( F ? ) reduce to the corresponding problem for the algebraic group G(F 9 ). Now it is time to tell you that the parameterization problem for G(¥q) was solved even before Steinberg's work by Chevalley, imitating the Lie-theoretic case G(C) mentioned above. Before discussing the character formula issue, let's consider how far we have come in discussing my reply to the philosophical issue, "Are all groups finite?" The first suggestion in my reply is that we cannot ignore G(C) and G(F 9 ), and I hope the initial history above of the parameterization for describing characteristic representations of G(F 9 ), through Steinberg and Chevalley, duplicating Cartan's "theory of the highest weight", is convincing evidence of the usefulness of looking at continuous and algebraic groups. The problems are easier for these more richly structured groups, and some have been solved. The second suggestion in my reply, that we can abstract from these continuous contexts, and perhaps go beyond them, is evidenced

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by what has happened to the parameterization theory since that time: First, Curtis and Richen [12], [13], [29] showed it was possible to carry out an analog of the parameterization process, suitably modified, in any finite group with an appropriate split BN pair. Second, Alperin [1] demonstrated with his celebrated conjecture (a main theme of our conference) that it was possible to formulate a version of the parameterization which makes sense for any finite group! Perhaps those involved in the Classification might also say that it was useful to know that corresponding simple group classifications already existed for G(C) and G(F q ), and the uniform theory of groups of Lie type which emerged was (and is) useful in efficiently dealing with many properties of known groups, and in formulating many general concepts. The structure in the Lie-theoretic case was not at all ignored, but, as above, it was only a starting point (together with involutions and the Odd Order Paper!) for a more general (and more elaborate) theory. Let's now go to the issue of a character formula for the describing characteristic representations of G(¥q). We are, I believe, far from a result as complete as the Classification, for irreducible modular representations of G(¥q). The few results we have put us at the beginning of the cycle, where it is still essential to learn from G(C) and G(F ? ). Nevertheless, it has been part of the point of view of myself and my colleagues, especially in CPS (Ed Cline, Brian Parshall, and myself), to develop a theory as purely algebraic as possible, to both try and attain a more general theory and to allow for elaborations diverging from the cleanest cases. This point of view is also important in our approach to proving the current main conjecture, due to Lusztig. Before describing it, let me describe one of CPS's main algebraic abstractions, which will at least make the Lusztig conjecture easier to explain. The discussion is largely borrowed from my exposition [32].

1.1 Highest weight categories, and examples. Fix a field k, and let e be an abelian k-category (that is, C is abelian, all Horn sets are ^-modules, and multiplications of morphisms is k-linear). In all cases we will consider here, C will simply be equivalent to the category of finite dimensional modules over a finite dimensional algebra, but it may not start out looking like that. We suppose the nonisomorphic irreducible objects L(k) to be indexed by the elements λ of a poset A, called weights. For simplicity we will assume A is finite here; for a more general notion (requiring only that the intervals of A be finite), the reader is referred to [7]. We will also assume for simplicity that

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all objects of G have finite length, that Horn sets between objects are finitedimensional over k, and, moreover, that Ende L(λ) = k for each λ e Λ. We assume that G has enough projectives, and let Ρ (λ) denote the projective cover of L(X). We say that e is a highest weight category if there are objects ν ( λ ) , λ e A, suchthat (1)

V(k) has head L(k), and all other composition factors of smaller weight than λ.

have

(2) There is an epimorphism Ρ (λ) —> V(X) with kernel filtered by objects ν ( μ ) with μ greater than λ. These conditions imply that is the largest epimorphic image of Ρ (λ) with λ maximal among the weights of its composition factors. Such objects arise naturally in Lie-theoretic contexts. We call V(k) a Weyl object, since it is a Weyl module in our favorite context of characteristic ρ algebraic group representations. Other good names are Verma object, or simply standard object. Typically, these objects are well understood, and the main object of research is to write the irreducible objects in terms of them in the Grothendieck group (that is, to obtain their "Weyl character formula"). This is precisely what the Lusztig conjecture purports to do, with certain restrictions.

Three examples in Lie theory. (1) The example which first motivated CPS is the following: Let G = G{k) be a semisimple, simply connected algebraic group over an algebraically closed field k of positive characteristic ρ (e.g. k = Fq ). This is the most relevant example for finite group theory. In discussing it, I will assume some basic terminology from algebraic group theory, but the reader familiar with the basic theory of root systems and Lie algebras as found in [20] should be able to follow much of it. Just keep in mind the basic example G — SL(n, F^). There, Τ below is the group of invertible diagonal matrices over ¥ q , Β is the group of upper triangular matrices and W is the group of η χ η permutation matrices. The Lie algebra of G as a vector space is η χ η matrices of trace 0, and its root spaces are just the 1 — dimensional spaces with arbitrary entries in the i, j position, for fixed i φ j, and 0 ' s elsewhere. These are common eigenspaces for the action of Τ by conjugation, and the associated homomorphism (character) mapping Τ to F* is called a root in the world of algebraic groups, while an arbitrary algebraic group homomorphism from Τ to ¥ q is called a weight. Let Γ be a fixed maximal torus, and denote the root system of Τ acting on the Lie algebra of G by Φ. We choose a

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set Φ + of positive roots, and let Β denote the corresponding Borel subgroup corresponding to the associated set Φ - of negative roots. The set X(T) of characters (weights) on Τ is partially ordered by the rule: λ < μ μ —λ = ηαα for non-negative integers ηα. We also have an induced poset structure on the set X(T)+ of dominant weights (relative to Φ + ) . Fix any finite set Λο of dominant weights, let Λ be the (finite) set of dominant weights λ for which λ < λο for some λο e Λο- Then the category C of finite-dimensional G— modules (in the sense of algebraic groups) which have composition factors each with maximal T— weight in A is a highest weight category with weight poset A. The Wey 1 modules V(X) are obtained as linear duals of modules induced to G, in the sense of algebraic groups, from dominant weights in X(T)+ extended to B. (These induced modules are all finitedimensional!) They may also be obtained by a reduction modulo ρ process from an irreducible module in characteristic 0. As such, their decomposition into weights for Τ is directly obtainable from the Weyl character formula. Projecting C onto any block of G -modules also gives a highest weight category. If ρ > h, the Coxeter number of the root system, it is well-known [22] that the character formulas for all irreducible modules are deducible from those in the principal block. The weights for the latter are the dominant weights in the orbit Wp.O of 0 under the 'dot' action of the affine Weyl group Wp (defined by w. μ = ιυ(μ + ρ) — ρ, where ρ is the sum of all the fundamental dominant weights, for w € Wp and μ € X(T).) Also, Steinberg's tensor product theorem allows us to restrict attention to restricted weights, those with coefficients less than ρ when expressed in terms of certain 'fundamental' weights. Let us redefine A as the set of dominant weights which are in the orbit Wp.0 and bounded above by a restricted weight in that orbit. Lusztig's conjecture may then be written chL(w.O) =

Σ

(-^w)~ay)Pywo,wwoWchV(y.O),

y. OeA

for any weight W.O in A. Here y, W are in Wp and WQ denotes the long word in the ordinary Weyl group W. The terms Fyujo.ioujoO), are values at 1 of Kazhdan-Lusztig polynomials, which are defined in a purely combinatorial way for any pair of elements in a Coxeter group. Finally i ( w ) denotes the length of w in the sense of Coxeter groups (the number of fundamental reflections in a minimal expression), and the function ch(—) just assigns an object of G to the associated element in the Grothendieck group of C. The conjectured formula would give character formulas for all irreducible G-modules, so long as ρ > 2h. — 3, and these would in turn give corresponding

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character formulas for any finite group G(F ? ) of Lie type associated to G, with q a power of p. (Actually, so long as ρ > h, the above formula could hold for all restricted weights, and as such would have the same implications for finite groups, for such a p. This stronger version of Lusztig's conjecture was formulated by Kato [23], who apparently originally believed it to be a consequence of the original conjecture, but the arithmetic doesn't work out that way. Let me take this opportunity to mention that the first open cases for the Lusztig conjecture occur for SL(5, 5) and SL(5, 7), which I have been examining with the help of an NSF undergraduate REU student. The former case is the first possibility for the Kato and Lusztig versions to diverge.) Lusztig obtained his conjecture by analogy with his conjecture with Kazhdan [24] for complex Lie algebras, which we describe next. (2) Let g be a complex semisimple Lie algebra, and fix a Cartan subalgebra f) and Borel subalgebra b containing I). Consider the corresponding category Ο of BGG. The objects are the g-modules which are f)-diagonalizable with finite-dimensional weight spaces (where a 'weight' here is a 1-dimensional representation for the Lie algebra f)), and with the set of nonzero weights bounded above by some finite set of weights. We will also restrict attention to the case where all weights are integral·, equivalently, they belong (by identification) to the set X(T) of characters for a torus Τ associated to I); these are just the integral linear combinations of the 'fundamental' weights for t). It is again true that any block of such modules forms a highest weight category, and all character formulas for irreducible modules are obtainable from the principal block case. The standard objects this time are the Verma modules M\, λ € X(T), obtained by tensor induction of λ at the universal enveloping algebra level from b to g. We write V(X) = Μχ, and let L(k) denote the irreducible head of ν ( λ ) . The weights A indexing irreducible modules in the principal block are just those in the orbit A — W. —2p. (This is also the orbit of 0 under the 'dot' action, since 0 = wq. —2p.) They correspond bijectively to elements of the Weyl group. The Kazhdan-Lusztig conjecture (now a theorem due to Brylinski-Kashiwara [6] and Beilinson-Bernstein [5]) reads ch L(w. - 2 ρ ) = Σ (~Vi{w)~t{y)Py,w{

1) ch V(y. -

2p),

y€W

where again Py,w(l) is the value at 1 of a Kazhdan-Lusztig polynomial. The similarity of this formula and the previous one is remarkable, and all the more so when one considers that the standard modules in the first case are finite-dimensional, but infinite-dimensional here. The next case, is even more

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remarkable, in that we obtain precisely the same character formula for standard objects which are not modules at all, but complexes of sheaves. (3) A key ingredient in the proof of the Kazhdan-Lusztig conjecture was the Kazhdan-Lusztig formula for the stalk dimensions of the cohomology of perverse sheaves. It can be written as a character formula in the Grothendieck group sense we are using here, and we describe it below. Let X = G/Β denote the flag variety obtained from the simply connected semisimple complex Lie group G associated to the Lie algebra g above, and consider the category G of perverse sheaves on X with respect to the Schubert stratification and the middle perversity [4]. (Thus a stratum is a Schubert cell S(w) = B w B / B , w G W, and a perverse sheaf is a complex of sheaves of complex vector spaces with cohomology locally constant (thus constant) and finite-dimensional on Schubert cells, with certain support conditions satisfied.) Theposetis W, with its Bruhat-Chevalley order, and the Weyl objects V(w), w € W, are quite easy to describe: V(w) = ^ ( ^ ( ^ ( ι υ ) ] , the extension by 0 of the constant sheaf, shifted downward as a complex in the derived category by degree £(w) Every Weyl object V(w) has a unique irreducible quotient L(w), and the axioms for a highest weight category are satisfied [28, §5]. Though unnecessary in our discussion, it is a remarkable fact that L(w) is the downward shift by £(w) of the complex (extended by zero to X ) defining Goreski-MacPherson intersection cohomology on the closure of S(w)\ see [25] and [33], The Grothendieck group formula of Kazhdan-Lusztig [25] reads ch L(w.

— 2p) = ^ ( - l ) i ( M , ) - £ W P y i I i ; ( l ) c h V ( y . - 2 p ) , yeW

which is identical to the form of the Verma module Kazhdan-Lusztig conjecture above. Essentially, the latter conjecture was proved through an equivalence of categories reducing it to the above formula. 1.2 Quasihereditary algebras. Every highest weight category with finite weight poset and all objects of finite length is the category of finite-dimensional modules for a quasihereditary algebra S. Indeed, CPS introduced quasihereditary algebras for this reason, and proved that, conversely, the category of modules for a quasihereditary algebra could be viewed as a highest weight category [31], [28], [7]. We will not reproduce the axioms for a quasihereditary algebra here, but note that examples include hereditary algebras and poset algebras [28], as well as all finite-dimensional algebras of global dimension

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two [17]. All quotient algebras of hereditary algebras are quasihereditary. Further Lie-theoretic examples of quasihereditary algebras include Schur algebras and q-Schur algebras, and their generalizations [7], [18]. CPS believes that understanding these algebras (and variations, with various degrees of added structure) will provide a good basis for understanding representations of algebraic groups in characteristic p, and finite groups of Lie type in describing or nondescribing characteristic. A new generalization of the quasihereditary notion, very relevant to the nondescribing characteristic case, is described in the last section of this paper. Every quasihereditary algebra S has finite global dimension. The opposite algebra Sop is quasihereditary with the same weight poset, and the 5-module Λ (λ) dual to the Weyl module Vop(X) for Sop has the following remarkable property [7; p. 98, bottom] = Ext"(V(/z),A(X)) = { * 10 otherwise. Here we have assumed, as before, that End L(X) — k to simplify the statement. Using this property, and an Euler characteristic argument of Delorme, it is possible to understand why the character formulas in each of the above three cases have such a remarkably similar appearance. Moreover, by tracking in the abstract setting a version (due to MacPherson, see [33]) of the arguments used to prove the Kazhdan-Lusztig formula in the perverse sheaf case, CPS was able to provide [8] (see also [9] and [10]) the following reductions. The 'length' ί{λ) of a weight λ = w.O below is the number of simple reflections in a reduced expression for w. Theorem (The CPS reductions). In each of the three examples above, the Lusztig conjecture or its analog (the Kazhdan-Lusztig conjecture, or KazhdanLusztig formula) is equivalent to each of the following statements:

(1) For each λ, μ e Λ, E x t ^ V ^ ) , Ζ,(λ)) φ 0 =>· ί{λ) - £(μ) = 1 (mod 2).

(2) For each λ € Λ, and each weight λ' adjacent to λ (in the sense that the affine Weyl group or Weyl group element associated to λ' is obtained from that associated to λ by right multiplication by a simple reflection), we have E x t 1 ^ ^ ' ) , Ζ,(λ)) φ 0. (By a duality principle, one may take here λ' < λ. or λ' > λ). (3) For each λ, μ e Λ, the natural map Ext^LOz), is surjective.

L(X))

Extl(V(ß),

L(k))

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The most promising of these reductions is perhaps the second one, though each has its own advantages. When the Lusztig conjecture is true, versions of 1) and 3) hold with the number 1 replaced by η throughout (two replacements in 1)), and dual versions hold using Α(μ), cf. [8]; see also [9]. One may study such conditions purely from the point of view of finite dimensional quasihereditary algebras, though they are far from giving us a description up to isomorphism (or a suitable weaker invariant) of the algebras involved. Until we get that far, we are somewhat in the position of trying to prove deep properties of finite simple groups without knowing what all the simple groups are.

1.3 The Lusztig program. Recently, George Lusztig [27] has formulated an attack on his own conjecture organized around the theory of quantum groups. Work by himself and Kazhdan relates a Lusztig-type conjecture for quantum groups at a root of unity to its validity in a category of 'negative level' representations for affine Kac-Moody Lie algebras. The latter conjecture, at least for simply laced root systems, is settled by work of Kashiwara and Tanisaki (also claimed by Casian, who acknowledges his original proof was in error) by reduction to a category of perverse sheaves on a generalized flag variety. Ignoring the non simply-laced case difficulties1, to complete the chain, one requires a reduction from quantum groups at a ptb. root of unity to algebraic groups in characteristic p. This has been provided for all types by Anderson, Jantzen and Soergel [3] for ρ sufficiently large, depending on the individual root system and its rank. Unfortunately, no specific bound whatsoever is known for ρ as of this writing. The problem is that ρ must stay away from divisors of the index in a maximal order of a certain algebra over Z, and the algebra is constructed so indirectly that very little information on the index is available. As Jantzen himself reported at the 1994 Banff conference, this situation is simply not acceptable to finite group theory. While CPS thinks highly of the AJS work, we regard the Lusztig conjecture as open and continue to work on it.

1.4 Stratified algebras. Another aspect of the situation is that CPS wants a theory sufficiently general to be appropriate for nondescribing characteristic, in the spirit of the Dipper-James work on the q-Schur algebra. Already there has been work by Dipper and others (see [19]) dealing with groups other than 1 These difficulties have apparently now been handled by Kashiwara-Tanisaki.

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the general linear group. While it may be that quasihereditary algebras are involved in these cases, at least in favorable characteristics, CPS conjectures a role for a slightly more general kind of algebra. This new generalization is called a stratified algebra [11]. I describe first the most basic types of stratified algebras, the algebras with a standard stratification, which are quite close to quasihereditary algebras, and one weaker notion, algebras (whose module category is) equipped with a stratifying system. The description is quite easy to do in both cases if we just think about how we are to relax the corresponding notion (1.1) of a highest weight category: First, we relax the condition that the weights Λ form a poset, requiring only that they form a quasiposet, so that two weights λ and μ may satisfy λ < μ and μ < λ without being equal. The equivalence classes thus obtained do themselves naturally form a poset A, and we let λ denote the element of Λ associated with λ e A. Next, in condition 1) for a highest weight category, we require only that all composition factors L(ß) of the standard object V(k) satisfy μ < λ. (So that L(X) may appear twice or more, along with other composition factors Ζ,(μ) with λ = μ.) The second condition 2) is kept in the "standard" case, and that completes the definition for that case. In the "stratifying system" case the inequality in 2) is relaxed to allow equality, but other relaxations are made as well: We just assume we have a system of objects and given projective objects Ρ (λ) mapping onto each of these with kernel filtered by V(^)'s with μ > λ. We do not require that V(k) have an irreducible head, or that the quasiposet index the irreducible modules. As a replacement for the latter, we do insist that every irreducible module appear in the head of some V(k). Condition 1) is replaced by the requirement that there are no nonzero homomorphisms from Ρ (λ) to ν(μ) unless μ < λ. In either case, one can prove from these conditions that the underlying algebra A has a sequence of idempotent ideals 0 = JQ C J\ C · · · C JN, with η = |A|, such than Ext™ /y .(M, N) = E x t Ν ) for all left (or right!) A/J—modules Μ, N, all integers η > 0, and each index i. With mention of A omitted, this is the general CPS notion of a stratified algebra, with a stratification of length«. In the "standard" case, each is left projective as an Λ/7, _i -module, and this is characteristic of standardly stratified algebras. The ideal need not be right projective, however. Unlike the quasihereditary case, there are algebras which are standardly left stratified but not standardly right stratified (even though the "general" notion of a stratified algebra is leftright symmetric). Apart from that, and the relaxed ordering inside standard modules, the theory of standardly stratified algebras is very close to that of

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quasihereditary algebras and highest weight categories. Note that the standard module ν(λ) is uniquely determined, in the "standard" case, as the largest quotient of Ρ (λ) with all composition factors Ζ-(μ,) satisfying μ < λ. The notion of an algebra whose (left) module category has a stratifying system, and the general notion of a stratified algebra, are both weaker, but more flexible. For instance, an algebra which has a stratification of length η > 1 in the "general" sense also has one of length η — 1, obtained by removing any of the intermediate ideals in the above chain. (There is an analog of this statement for algebras with a stratifying system.) Also, as mentioned, the "general" notion is left-right symmetric, though this is not obvious. CPS has taken considerable trouble [11] to be able to recognize when an endomorphism algebra (e.g. a Schur algebra, g-Schur algebra, or future generalization) has a natural structure as a stratified algebra. This includes, of course, the quasihereditary case (easy to check in the presence of a standard stratification), but the generalizations are also interesting, and may be important. The recognition conditions are quite complicated, though simplify very considerably 2 in special cases. They involve a kind of generalized "Specht module" theory. Rather than reproduce the conditions here, I will simply give some examples from [11], referring the reader to that paper for additional details, and further examples: Throughout k is an algebraically closed field. (1) A stratification of length 2. Let G be any nontrivial finite group. Consider the direct sum Τ of the trivial module and the regular permutation module over k. Then A = End*G(O is stratified, with |A| = 2 (the length of the stratification). Interesting cases occur already for G the cyclic group of order 2 or the Klein four group. In the first case, A is the wellknown algebra of dimension 5 which has two simple modules a and b, a b with projective covers b and . In the second case, A is already not a quasihereditary! Its projective covers have Loewy layers as indicated by a b the diagrams a a b and . a a 2 The simplifications in the preprint "Stratifying endomorphism algebras over Hecke algebras", by Du, Parshall, and Scott, over Z[q, q~x], might even be described as dramatic.

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Thus, unlike the quasihereditary case, the standard object associated to a (the quotient of the first projective cover by the submodule isomorphic to the second) has a appearing with a nontrivial multiplicity. Nevertheless, inside A, the ideal generated by the idempotent associated to the second projective cover is quite nice. It is both idempotent and (left) projective. This example and the next are both standardly stratified. All standardly stratified algebras have such ideals, and their factor algebras are also standardly stratified. (2) A stratification of length 3: the dihedral group of order 8. Let G be the dihedral group of order 8, and let Τ be the direct sum of the trivial module, the regular module, and the two transitive permutation modules of degree 4 associated to coset spaces of the two conjugacy classes of noncentral subgroups of order 2. We take char k = 2. This time the algebra A = End^c Τ is quite difficult to visualize from the given data. The general CPS approach is to try to impose a generalized 'Specht module' filtration on T, and deduce from its properties that A is indeed stratified. Without giving full details, I will at least describe the 'Specht modules' we use. Let a and b denote generators of order two for G, and put Λ = a - 1, Β = b - 1. Thus A 2 = 0 = B2 and AB AB = Β ABA. We will diagram cyclic modules by indicating where a nontrivial action of A or Β occurs, starting from a generator. (No arrow associated to a given node and label indicates a zero multiplication. Note that, if a node was reached by multiplication by A, then that node must be killed by A. A parallel statement holds for B.) Thus, the four transitive permutation modules making up Τ have diagrams

A

Β A

/

\

•

i

i

•

•

;

I

•

•

Β

\

S

•

Β

•

A

A

I

•

Β

•

A Β

1,

•

A

•

Β

I 1 •

Β

A •

•

Here the single node · is also representative of the unique irreducible (trivial) module for the group G, and the above diagrams give refined Loewy series pictures. In the CPS set-up, each of these indecomposable components of Τ has a 'Specht filtration', which turns into the required filtration of projective covers by standard modules for the algebra A = End^cC^) under the contravariant

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functor Hom^cC—, T). (This functor is not exact, but a filtration of Τ still induces a filtration of A.) Moreover, each component above has a distinguished 'Specht' submodule, though it is possible for two such 'Specht' modules to be isomorphic. For the first and second components above, the Specht submodule is the 1-dimensional trivial module, while for the third and fourth component, the Specht submodule is the unique 3-dimensional submodule. The reader will observe that, when the bottom and top trivial modules are eliminated from the second component, the remainder is the direct sum of the Specht modules associated to the third and fourth component. This puts all 'Specht filtrations' in evidence. The reader may consult [11] for some general machinery to check that these filtrations are transformed into the required standard module filtrations for the indecomposable projective components of A (or can attempt a direct verification starting from the filtrations and functor we have given). CPS believes something general is happening here for all Coxeter groups, that a similar construction always leads to a nontrivial stratified algebra. We have conjectured the following:

Conjecture. Let W be a finite Coxeter group with distinguished generating set S, |S| > 1. For J c S, let Tj denote the permutation module for kW on the cosets {Wjw}wew- Put Τ = 0 y c 5 Tj and A = End^w T. Then A is stratified with respect to a quasiposet A with \ A | > 3.

For a more detailed statement of the conjecture, see [11]. We believe the stratification arises from a stratifying system, and that A may be assumed to have a largest and smallest element, containing only one element each as equivalence classes in A, with these elements associated to Ts (the trivial module) and the sign module submodule of Τφ. The stronger form of the conjecture also has as a consequence the existence of certain known resolutions, one of which is the Coxeter complex. As the final version of this paper is being readied for press, it appears that Du, Parshall and Scott will soon prove the stronger conjecture, and a ^-analog. 3 CPS also expects (with Jie Du) that an enlarged version of A will have a standard stratification related to the filtration of the Tj's by dual left cell modules, in the sense of Kazhdan-Lusztig, of length equal to the number of two-sided cells. The algebra A itself exhibits such a stratification in the order 8 dihedral case above. This is a special case of a Weyl group of type B. In 3 This has now been done, in the preprint described in the previous footnote.

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this case Jie Du and I are working on another natural enlargement that may be quasihereditary. 4 As mentioned above, it also seems likely that a ^-analog of the conjecture holds for Hecke algebras, a possibility which makes the conjecture and stratified algebras quite relevant to nondescribing characteristic theory. This already important area of research will become even more central once the problems posed by the Lusztig conjecture itself are solved.

References [1] [2] [3]

[4] [5]

J. Alperin, Cohomology in representation theory, Proc. Symp. Pure Math. 47 (1987). 3-11. M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Algebra 92(1985), 44-80. H. Andersen, J. Jantzen, and W. Soergel, Representations of quantum groups at a ρ th root of unity and of semisimple algebraic groups in characteristic p: Independence of p, Asterisque 220 (1994). A. Beilinson, J. Bernstein, and P. Deligne, Analyse et topologie sur les espaces singulares, Asterisque 100 (1982). A. Beilinson and J. Bernstein, Localisation des S-modules, C. R. Acad. Sei. Paris 292(1981), 15-18.

[6]

J. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387^10.

[7]

E. Cline, B. Parshall and L. Scott, Finite dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85-99.

[8]

E. Cline, B. Parshall and L. Scott, Abstract Kazhdan-Lusztig theories, Tohoku Math. 45 (1993), 511-534. E. Cline, B. Parshall and L. Scott, Infinitesimal Kazhdan-Lusztig theories, Contemp. Math. 39 (1992), 43-73. E. Cline, B. Parshall and L. Scott, Simulating perverse sheaves in modular representation theory, Proc. Symp. Pure Math. 56 (1994), 63-104.

[9] [10] [11] [12] [13]

E. Cline, B. Parshall and L. Scott, Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 591, Vol. 124, 1996. C. Curtis, Irreducible representations of finite groups of Lie type, J. Reine Angew. Math. 219 (1965), 180-199. C. Curtis, Modular representations of finite groups with a split (B,N) pair, Seminar on algebraic groups and related finite groups (A. Borel. ed.), Lecture Notes in Math. 131, Springer-Verlag, 1970, 57-95.

4 This is true. The preprint by Du and Scott is entitled "The