Group Representations Volume 1. Part A: Background Material + Part B: Introduction to Group Representations and Characters

1353 Pages·1992·61.121 MB·English

by  Gregory Karpilovsky

GROUP REPRESENTATIONS
Volume 1
Part A: Background Material
NORTH-HOLLAND MATHEMATICS STUDIES 175
{Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN
Centro Brasileiro de Pesquisas Fisicas
Rio de Janeiro, Brazil
and
University of Rochester
New York, U.S.A.
NORTH-HOLLAND - AMSTERDAM • LONDON • NEW YORK• TOKYO
GROUP REPRESENTATIONS
Volume 1
Part A: Background Material
Gregory KARPILOVSKY
Department of Mathematics
California State University
Chico, CA, U.S.A.
~
~
ffl
1992
NORTH-HOLLAND - AMSTERDAM • LONDON • NEW YORK• TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V.
Sara Burgerhartstraat 25
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
Library of Congress Cataloging-in-Publication Data
Karpilovsky, Gregory, 1940 -
Group representations/ Gregory Karpilovsky.
p. cm ... (North-Holland mathematics studies;l 75)
Includes bibliographical references and index.
Contents; v., pt .. A. Background material, pt. B. Introduction to
group representations and characters.
ISBN 0-444 -88632-X (set)
1. Representations of groups. I. Title. II. Series.
QAl 76.K37 1992 92-14786
512' .2-dc20 CIP
ISBN: 0 444 88632 X (Set: Part A and B)
© 1992 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved.
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This book is printed on acid-free paper.
Printed in The Netherlands
To my wife Helen,
who typed this lengthy manuscript
and remained cheerful throughout

vii
Preface
The present volume is the first of a multi-volume treatise on group repre
sentations. The principal object of these volumes is to provide, in a self
contained manner, a comprehensive coverage of the mainstream of group
representation theory. The appropriate audience for these volumes consists
of aspiring graduate students and mature mathematicians working in the
field of group representations. No mathematical knowledge is presupposed
beyond the rudiments of abstract algebra, set theory and field theory; how
ever, a certain maturity in mathematical reasoning is required. Apart from
a few obvious exceptions ( e.g. odd order theorem, classification of finite sim
ple groups, etc.), the volumes are entirely self-contained. The style of the
presentation is informal : the author is not afraid to repeat definitions and
formulas when they are needed. Many sections begin with a nontechnical
description of what is about to be done. A special effort has been made to
render the exposition transparent.
Due to bookbinding considerations, Volume 1 is split into two parts, Part
A and Part B. However, mathematically, the material is divided into three
parts. The first part provides background material for all volumes. The
second part is devoted to introduction to group representations, and the
third part to introduction to group characters. Many of the results are pre
sented in their greatest generality and much of the material discussed is not
conveniently available in other monographs on the subject. A systematic
description of the material is supplied by the introductions to individual
chapters and therefore will not be repeated here.
A word about notation. As is customary, Theorem 2.3.4 denotes the
fourth result in Section 3 of Chapter 2; however, for simplicity, all references
to this result within Chapter 2 itself, are designated as Theorem 3.4.
I am indebted to my wife Helen, who not only typed the entire volume,
but encouraged me (always with patience and good humour) throughout the
entire project.
California State University, Chico G. Karpilovsky
November, 1991

Contents
Preface
Vll
Part I Background Material 1
1. Rings and Modules 3
1.1. Notation and terminology 3
1.2. Preliminary results 8
1.3. Artinian and noetherian modules and rings 13
1.4. Semisimple modules 22
1.5. The radical and socle of modules and rings 26
1.5.A. The radical and socle of modules 26
1.5.B. The Jacobson radical 31
1.5.C. The Jacobson radical and idempotents 39
1.6. Idempotent lifting theory 42
1.6.A. General results 42
1.6 .B. Semiregular rings 46
1.6.C. Algebras over complete rings 52
1.7. Azumaya's theorems 55
1.8. Local rings 60
1.9. Endomorphism algebras 61
1.10. Strongly indecomposable modules 65
1.10.A. Basic properties and characterizations 65
1.10.B. Azumaya's decomposition theorem 70
1.10.C. The Krull-Schmidt theorem 71
1.11. Direct decompositions and blocks 73
1.12. Matrix rings 80
2. Artinian and Semilocal Rings 89
2.1. Semiprimitive artinian rings 89
2.2. Semilocal rings 92
2.3. Artinian rings 100
2.3.A. A characterization 100