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Lecture Notes in Mathematics 1081 Lecture Notes in Mathematics 1081 Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris David J. Benson Modular Representation Theory New Trends and Methods Second printing Sprin ger Author David J. Benson Department of Mathematical Sciences University of Aberdeen Meston Building King's College Aberdeen AB24 SUE Scotland UK Modular Representation Theory Library of Congress Cataloging in Publication Data. Benson, David, 1955-. Modular representation theory. (Lecture notes in mathematics; 1081) Bibliography: p. Includes index. 1. Modular representations of groups. 2. Rings (Algebra) I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1081. QA3.L28 no. 1081 [QA171] 510s [512'.2] 84-20207 ISBN 0-387-13389-5 (U.S.) Mathematics Subject Classification (1980): 20C20 Second printing 2006 ISSN 0075-8434 ISBN-10 3-540-13389-5 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-13389-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science-i-Business Media springer.com © Springer-Verlag Berlin Heidelberg 1984 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production: LE-T^X Jelonek, Schmidt & Vockler GbR, Leipzig Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11749158 41/3100/YL 5 4 3 2 10 Introduction This book grew out of a graduate course which I gave at Yale University in the spring semester of 1983. The aim of this course was to make some recent results in modular representation theory accessible to an audience ranging from second-year graduate students to established mathematicians. The material covered has remarkably little overlap with the material currently available in textbook form. The reader new to modular representation theory is therefore encouraged also to read, for example, Feit [51], Curtis and Reiner [37,38], Dornhoff [44], Landrock [65], as well as Brauer*s collected works [16], for rather different angles on the subject. The first of the book's two chapters is intended as background material from the theory of rings and modules. The reader is expected already to be familiar with a large proportion of this, and to refer to the rest as he needs it; proofs are included for the sake of completeness. The second chapter treats three main topics in detail. (i) Representation rings. (ii) Almost split sequences and the Auslander-Reiten quiver. (iii) Complexity and cohomology varieties. 1 hope to impress upon the reader that these three topics are closely connected, and to encourage further investigation of their interplay. The study of modular representation theory was in some sense started by L. E. Dickson [40] in 1902. However, it was not until R. Brauer [16] started investigating the subject that it really got off the ground. In the years between 1935 and his death in 1977, he almost single-handedly constructed the corpus of what is now regarded as the classical modular representation theory. Brauer's main motivation in studying modular representations was to obtain number theoretic restrictions on the possible behaviour of ordinary character tables, and thereby find restrictions upon the structure of finite groups. His work has been a major tool in the classification of the finite simple groups. For a definitive account of modular representation theory from the Brauer viewpoint (as well as some more modern material) see Feit [51]. It was really J. A. Green who first systematically developed the study of modular representation theory from the point of view of IV examining the set of indecomposable modules, starting with his paper [54]. Green's results were an indispensable tool in the treatment by- Thompson, and then more fully by Dade, of blocks with cyclic defect groups. Since then, many other people have become interested in the study of the modules for their own sake. In the study of representation theory in characteristic zero, it is customary to work in terms of the character table, namely the square table whose rows are indexed by the ordinary irreducible representations, whose columns are indexed by the conjugacy classes of group elements, and where a typical entry gives the trace of the group element on the representation. Why do we use the trace function? This is because the maps V ^»- tr(g,V) are precisely the algebra homomorphisms from the representation ring to (E, and these homomorphisms separate representa­ tions. In particular, in this case the representation ring is semi- simple. This has the effect that we can compute with representations easily and effectively in terms of their characters; representations are distinguished by their characters, direct sum corresponds to addition and tensor product corresponds to multiplication. The orthogonality relations state that we may determine the dimension of the space of homomorphisms from one representation to another by taking the inner product of their characters. How much of this carries over to characteristic p, where p||G| ? The first problem is that Maschke's theorem no longer holds; a representation may be indecomposable without being irreducible. Thus the concepts of representation ring A(G) and Grothendieck ring do not coincide. The latter is a quotient of the former by the "ideal of short exact sequences" A (G,l). Brauer discovered the remarkable fact that the Grothendieck ring A(G)/A (G,l) is semisimple, and found the set of algebra homomorphisms from this to (E, in terms of lifting eigenvalues. Thus he gets a square character table, giving information about composition factors of modules, but saying nothing about how they are glued together. In an attempt to generalize this, we define a species of the representation ring to be an algebra homomorphism A(G) -• (E. Even if we use the set of all species, we cannot distinguish between modules Vi and V2 when V-j^ - V2 is nilpotent as an element of A(G) . For some time, it was conjectured that A(G) has no nilpotent elements in general. However, it is now known that A(G) has no nilpotent elements whenever kG has finite representation type (i.e. the Sylow p-subgroups of G are cyclic, where p = char(k)), as well as a few other cases in characteristic two, whilst in general there are nilpotents (see O'Reilly [98] and Zemanek [95, 96] as well as a forthcoming paper by J. Carlson and the author). The Brauer species (i.e. the species of A(G) which vanish on AQ(G,1)) may be evaluated by first restricting down to a cyclic sub­ group of order coprime to p, and then lifting eigenvalues. The corresponding concept for a general species is the origin, namely the minimal subgroup through which the species factors. We show that the origins of a species are very restricted in shape, namely if H is an origin then H/0 (H) is a cyclic group of order coprime to p, and we show how 0 (H) is related to the vertices of modules on which the species does not vanish. \ Many of the properties of representation rings and species are governed by the trivial source subring A(G,Triv), which is a finite dimensional semisimple subring of A(G). Thus we spend several sections investigating trivial source modules, and showing how these modules are connected with block theory. Instead of developing defect groups and Brauer*s first main theorem just for group algebras, we develop them for arbitrary permutation modules, and recover the classical case by applying the theory to G x G acting on the set of elements of G by left and right multiplication. When applied to this case, the orthogonality relations 2.6.4 become the ordinary character orthogonality relations. In ordinary character theory, one of the ways in which the structure of the group is reflected in the character table is via the so-called power maps, or Adams operations. Namely there are ring homomorphisms \|r^ on the character ring, with the property that the character value of g on t^(V) is the character value of g^ on V. These are usually given in terms of the exterior power operations A , and these operations make the character ring into a special lambda-ring. It turns out that for modular representations we must first construct the ring homomorphisms \Jr^: a(G) -• a(G) , and then use them to construct operations \^, which do not agree with the exterior power operations unless n < p (although they do at the level of Brauer characters) , and the \^ make a(G) S> 2'[l/p] into a special 2 lambda-ring. It then makes sense to use the t to define the powers of a species. As an application of these power maps, we give Kervaire's proof that the determinant of the Cartan matrix is a power of p, rather than using Brauer's characterization of characters. VI The next feature of ordinary character theory which we may wish to mimic is the fact that the orthogonality relations may be interpreted as saying that a module is characterized by its inner products with the indecomposable modules.
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