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Prerequisites in Module Theory Modular Representation Theory of Finite Groups Peter Schneider Modular Representation Theory of Finite Groups Peter Schneider Department of Mathematics University of Münster Münster Germany ISBN 978-1-4471-4831-9 ISBN 978-1-4471-4832-6 (eBook) DOI 10.1007/978-1-4471-4832-6 Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2012954001 Mathematics Subject Classification: 20C20, 20C05 © Springer-Verlag London 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The nature of the representation theory of a finite group G in (finite-dimensional) vector spaces over some field k depends very much on the relation between the order |G| of the group G and the characteristic char(k) of the field k. If char(k) does not divide |G| then all representations are semisimple, i.e. are direct sums of irreducible representations. The reason for this is the semisimplicity of the group al- gebra k[G] in this situation. By the modular representation theory of G one means, on the other hand, the case where char(k) is a divisor of |G| (so that, in particular, char(k) must be a prime number). The group algebra k[G] now may be far from be- ing semisimple. In the extreme case, for example, where |G| is a power of char(k), it is a local ring; there is then a single irreducible representation, which is the trivial one, whereas the structure of a general representation will still be very complicated. As a consequence a whole range of additional tools have to be developed and used in the course of the investigation. To mention some, there is the systematic use of Grothendieck groups (Chap. 2) as well as Green’s direct analysis of indecompos- able representations (Chap. 4). There also is the strategy of writing the category of all k[G]-modules as the direct product of certain subcategories, the so-called blocks of G, by using the action of the primitive idempotents in the center of k[G]. Brauer’s approach then establishes correspondences between the blocks of G and blocks of certain subgroups of G (Chap. 5), the philosophy being that one is thereby reduced to a simpler situation. This allows us, in particular, to measure how nonsemisimple a category a block is by the size and structure of its so-called defect group. Begin- ning in Sect. 4.4 all these concepts are made explicit for the example of the group G = SL2(Fp). The present book is to be thought of as an introduction to the major tools and strategies of modular representation theory. Its content was taught during a course lasting the full academic year 2010/2011 at Münster. Some basic algebra together with the semisimple case were assumed to be known, although all facts to be used are restated (without proofs) in the text. Otherwise the book is entirely self- contained. The references [1–10] provide a complete list of the sources I have drawn upon. Of course, there already exist several textbooks on the subject. The older ones like [5] and [6] are written in a mostly group theoretic language. The beautiful v vi Preface book [1] develops the theory entirely from the module theoretic point of view but leaves out completely the comparison with group theoretic concepts. For example, the concept of defect groups can be introduced either purely group theoretically or purely module theoretically. To my knowledge all existing books essentially restrict themselves to a discussion of one of these approaches only. Although my presenta- tion is strongly biased towards the module theoretic point of view, I make an attempt to strike a certain balance by also showing the reader the other aspect. In particular, in the case of defect groups a detailed proof of the equivalence of the two approaches will be given. This book is not addressed to experts. It does not discuss any very advanced aspects nor any specialized results of the theory. The aim is to familiarize students at the masters level with the basic results, tools, and techniques of a beautiful and important algebraic theory, hopefully enabling them to subsequently pursue their own more specialized problems. I wish to thank T. Schmidt for carefully reading a first draft and I. Reckermann and G. Dierkes for their excellent typesetting of the manuscript. Münster, Germany Peter Schneider Contents 1 Prerequisites in Module Theory ..................... 1 1.1ChainConditionsandMore..................... 1 1.2Radicals............................... 3 1.3 I -Adic Completeness ........................ 4 1.4 Unique Decomposition ....................... 9 1.5 Idempotents and Blocks ....................... 15 1.6 Projective Modules ......................... 26 1.7 Grothendieck Groups ........................ 34 2 The Cartan–Brauer Triangle ...................... 43 2.1 The Setting . ............................. 43 2.2 The Triangle ............................. 46 2.3 The Ring Structure of RF (G), and Induction . ......... 54 2.4TheBurnsideRing.......................... 59 2.5 Clifford Theory ........................... 67 2.6 Brauer’s Induction Theorem . ................... 71 2.7 Splitting Fields ............................ 75 2.8 Properties of the Cartan–Brauer Triangle .............. 78 3 The Brauer Character .......................... 87 3.1Definitions.............................. 87 3.2 Properties . ............................. 91 4 Green’s Theory of Indecomposable Modules .............. 97 4.1 Relatively Projective Modules . ................... 97 4.2 Vertices and Sources .........................105 4.3 The Green Correspondence . ...................110 4.4 An Example: The Group SL2(Fp) ..................119 4.5 Green’s Indecomposability Theorem ................140 5 Blocks ...................................147 5.1 Blocks and Simple Modules . ...................147 vii viii Contents 5.2CentralCharacters..........................151 5.3 Defect Groups ............................153 5.4 The Brauer Correspondence . ...................159 5.5 Brauer Homomorphisms .......................165 References ...................................175 Index ......................................177 Chapter 1 Prerequisites in Module Theory Let R be an arbitrary (not necessarily commutative) ring (with unit). By an R-module we will always mean a left R-module. All ring homomorphisms respect the unit element, but a subring may have a different unit element. 1.1 Chain Conditions and More For an R-module M we have the notions of being finitely generated, artinian, noetherian, simple, and semisimple. The ring R is called left artinian, resp. left noetherian, resp. semisimple, if it has this property as a left module over itself. Proposition 1.1.1 i. The R-module M is noetherian if and only if any submodule of M is finitely generated. ii. Let L ⊆ M be a submodule; then M is artinian, resp. noetherian, if and only if L and M/L are artinian, resp. noetherian. iii. If R is left artinian, resp. left noetherian, then every finitely generated R-module M is artinian, resp. noetherian. iv. If R is left noetherian then an R-module M is noetherian if and only if it is finitely generated. Proposition 1.1.2 (Jordan–Hölder) For any R-module M the following conditions are equivalent: i. M is artinian and noetherian; ii. M has a composition series {0}=M0 ⊆ M1 ⊆ ··· ⊆ Mn = M such that all Mi/Mi−1 are simple R-modules. P. Schneider, Modular Representation Theory of Finite Groups, 1 DOI 10.1007/978-1-4471-4832-6_1, © Springer-Verlag London 2013 2 1 Prerequisites in Module Theory In this case two composition series {0}=M0 ⊆ M1 ⊆···⊆Mn = M and {0}= ∼ L0 ⊆ L1 ⊆···⊆Lm = M satisfy n = m and Li/Li−1 = Mσ(i)/Mσ(i)−1, for any 1 ≤ i ≤ m, where σ is an appropriate permutation of {1,...,n}. An R-module M which satisfies the conditions of Proposition 1.1.2 is called of finite length and the integer l(M) := n is its length.Let Rˆ := set of all isomorphism classes of simple R-modules. For τ ∈ Rˆ and an R-module M the τ -isotypic component of M is M(τ):= sum of all simple submodules of M in τ . Lemma 1.1.3 For any R-module homomorphism f : L −→ M we have f(L(τ))⊆ M(τ). Proposition 1.1.4 i. For any R-module M the following conditions are equivalent: a. M is semisimple, i.e. isomorphic to a direct sum of simple R-modules; b. M is the sum of its simple submodules; c.
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