Table Of ContentSymmetric and G-algebras
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science. Amsterdam. The Netherlands
Editorial Board:
F. CALOGERO, Universita degJi Studi di Roma, Italy
Yu. I. MANIN, StelcJov Institute ofM athematics. Moscow. U.S.S.R.
A. H. G. RINNOOY KAN, Erasmus University. Rotterdam. The Netherlands
G.-C. ROTA, Ml.T., Cambridge, Mass., U.SA.
Volume 60
Symmetric and G-algebras
With Applications to Group Representations
by
Gregory Karpilovsky
Department of Mathematics,
California State University, Chico,
U.S.A.
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging in Publication Data
Karpi lovsky. Gregory. 1940-
Symmetrlc and G-algebras with applications to group
representations I Gregory Karpilovsky.
p. cm. -- (Mathematics and its applications; v.60)
ISBN-13:978-94-010-6756-0 e-ISBN-13:978-94-009-0597-9
DOl: 10.1007/978-94-009-0597-9
1. Frobenius algebras. 2. Representations of groups. 3. Group
algebras. I. Title. II. Title: G-algebras. III. Series:
Mathematics and its applications (Kluwer Academic Publishers) ; v
60.
OA251.5.K36 1990
512' .24--dc20 90-34936
CIP
ISBN-I3:978-94-010-6756-0
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All Rights Reserved
© 1990 by Kluwer Academic Publishers
Softcover reprint of the hardcover 1st edition 200 1
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To my children
SUZANNE and ELLOITT
SERIES EDITOR'S PREFACE
'Et moi •...• si j'avait su comment en revenir, One service mathematics has rendered the
je n'y serais point aIle.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non
The series is divergent; therefore we may be sense'.
able to do something with it. Eric T. Bell
o. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. it
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields. crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics'. 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large. all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes). and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And.
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
viii SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
Symmetry is important. It does not necessarily involve groups, and can take many forms. And, if
the past two centuries are anything to go by, the concept and its generalizations will become even
more important.
In algebra 'symmetry' manifests itself for instance in the theory of Frobenius algebras, sym
metric algebras and G-algebras. The theory of these has seen rapid growth in the last 15 years or so
and has reached a level where a first systematic and comprehensive account becomes both possible
and desirable. Moreover, there are important applications, particularly to group representation
theory as the present book amply shows.
The shortest path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them: the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
I'o ccasion de fesoudre des problemes ... eIle right than other people, but to be wrong for
nous fait pressentir la solution. more sophisticated reasons.
H. Poincare David Butler
Amsterdam, 25 April 1990 Michiel Hazewinkel
Preface
In this volume we present the theory of symmetric and G-algebras to
gether with a number of applications to modular group representation
theory and block theory of group algebras. The theory of symmetric
and G-algebras has experienced a rapid growth in the last ten to fifteen
years, acquiring mathematical depth and significance and leading to
new insights in group representation theory. By now a stage has clearly
been reached when a systematic account has become both possible and
desirable. We hope to help the reader early on to an understanding of
the basic structure of the theory, and to motivate at each successive
stage the introduction of new concepts and new tools.
This book is written on the assumption that the reader has had the
equivalent of a standard first-year graduate algebra course. Apart from
a few specific results and the general knowledge we have presupposed,
the book is entirely self-contained. A systematic description of the ma
terial is supplied by the introduction to individual chapters. There is a
fairly large bibliography of works which are either directly relevant to
the text or offer supplementary material of interest. An attempt has
been made to give credit for some of the major methods and theorems,
but we have stopped far short of trying to trace each theorem to its
source.
A word about notation. As is customary, Theorem 2.4.2 denotes
the second result of Section 4 of Chapter 2; however, for simplicity,
all references to this result within Chapter 2 itself, are designated as
Theorem 4.2.
The following is a brief description of the content of the book. In
Chapter 1 we review some basic ring-theoretic results. Many readers
may wish to glance briefly at the contents of this chapter, referring back
to the relevant sections when they are needed later.
In Chapter 2 we provide an extensive account of the theory of Frobe
nius and symmetric algebras. After recording some basic facts, we show
that if A is a Frobenius algebra over a field F and G a finite group, then
any crossed product F-algebra A*G is Frobenius. We then establish an
important result due to Harris (1988) which asserts that if A is semisim
*
pIe, then any crossed product F-algebra A G is symmetric. Turning
to modules over group algebras, we apply the above results to demon-
ix
x
strate that if V is a finitely generated semisimple F N -module (N is a
normal subgroup of G), then the F-algebra EndFG(VG) is symmetric.
As a byproduct of our investigations, we establish a number of impor
tant results pertaining to modular representation theory of groups and
block theory of group algebras.
The aim of Chapter 3 is twofold: first to investigate symmetric lo
cal algebras A over an algebraically closed field F with dimFZ(A) < 5
and second to apply the results obtained to modular representation
theory of finite groups. More specifically, assume that F is of prime
characteristic p and B is a block of a group algebra FG with exactly
one simple FG-module. Denote by k(B) the number of irreducible
complex characters in B and by D a defect group of B. As an applica
tion of our results on symmetric local algebras, we demonstrate that if
k(B) :::; 4, then k(B) = IDI and if k(B) = 5, then either IDI = 5 or D
is nonabelian of order 8. The results above are relevant to the Brauer's
conjecture which states that for any B, k(B) :::; IDI.
Chapter 4, the final chapter, provides an extensive account of the
theory of G-algebras. Much of the material presented is based on works
of Green (1968), Puig (1981), Broue and Robinson (1986) and Ikeda
(1987). As one of the many applications, we present a criterion for an
indecomposable two-sided ideal of RG to be a block of RG. Among
other applications, we mention a remarkable result due to Robinson
(1983, Corollary 1) regarding defect groups of blocks.
I would like to express my gratitude to my wonderful wife for the
tremendous help and constant encouragement which she has given me
in the preparation of this book. My grateful thanks to my daughter,
Suzanne, and my son Elliott, for bearing my great despair, forgiving
my inattentions, and denying me pity.
Contents
Preface ix
Notation xiii
Chapter 1. Preliminaries 1
1. Notation and terminology 1
2. Artinian, noetherian and semisimple modules 4
3. Semisimple modules 9
4. The radical and sode of modules and rings 13
5. The Krull-Schmidt theorem 22
6. Matrix rings 27
7. The Wedderburn-Artin theorem 34
8. Tensor products 37
9. Group algebras 41
Chapter 2. Frobenius and symmetric algebras 47
1. Definitions and elementary properties 48
2. Frobenius crossed products 56
3. Symmetric crossed products 61
4. Symmetric endomorphism algebras 72
5. Projective covers and injective hulls 81
6. Classical results 93
7. Frobenius uniserial algebras 109
8. Characterizations of Frobenius algebras 122
9. Characters of symmetric algebras 139
10. Applications to projective modular representations 147
11. Kiilshammer's theorems 165
12. Applications 180
