Group Representations : Volume 3

925 Pages·1994·13.67 MB·English

by  Gregory Karpilovsky

G R 0 U P REPR ES EN TATlO N S
Volume 3
NORTH-HOLLAND MATHEMATICS STUDIES 180
(Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN t
Centro Brasileiro de Pesquisas Fisicas
Rio de Janeiro, Brazil
and
University of Rochester
New York, U.S.A.
Leopoldo Nachbin passed away in April 1993. As Editor of the
Mathematics Studies, he will be succeeded by Saul Lubkin.
The present book was recommended for publication by Leopoldo Nachbin.
NORTH-HOLLAND- AMSTERDAM LONDON NEW YORK TOKYO
G RO UP REP R ES E NTATlO NS
Volume 3
Gregory KARPILOVSKY
Department of Mathematics
California State University
Chico, CA, USA
1994
NORTH-HOLLAND -AMSTERDAM LONDON NEW YORK TOKYO
ELSEVIER SCIENCE B.V.
Sara Burgerhartstraat 25
KO. Box 21 1, 1000 AE Amsterdam, The Netherlands
ISBN: 0 444 87433 X
0 1994 Elsevier Science B.V. All rights reserved.
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operation of any methods, products, instructions or ideas contained in the material herein.
This book is printed on acid-free paper.
Printed in the Netherlands.
To my wife
Helen
who typed it all
Everything in nature is lyrical in its ideal essence:
tragic in its fate, and comic in its existence.
George Santayana
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vii
Preface
The present book is the third volume of a multi-volume treatise on group
representations. Our principal goal is to provide, in a self-contained manner,
a comprehensive coverage of projective character theory and Clifford theory.
Certain topics concerning projective representations are also covered.
The book can be roughly divided into two parts, which will not preclude,
however, some strong interrelations between these. The first part is devoted
to the investigation of various properties of projective characters. Special
attention is drawn to spin representations and their character tables and
to various correspondences for projective characters. Among other topics,
we mention projective Schur index and projective representations of abelian
groups. The last topic is investigated by introducing a symplectic geometry
on finite abelian groups.
The second part is devoted to Clifford theory for graded algebras and its
application to the corresponding theory for group algebras. The volume ends
with a detailed investigation of the Schur index for ordinary representations.
A prominant role in our discussion is played by Brauer groups together with
cyclotomic algebras and cyclic algebras. The reader who wishes to obtain a
more detailed summary account of the contents of this volume, can have it
by reading through the brief introductions with which I begin each chapter.
A word about notation. As is customary, Theorem 5.3.4 denotes the
fourth result in Section 3 of Chapter 5; however, for simplicity, all references
to this result within Chapter 5 itself, are designated as Theorem 3.4.
I would like to express my gratitude to my wife for the tremendous help
and encouragement she has given me in the preparation of this book. For
answering specific quieries on topics contained in the text I am indebted to
J.-P. Serre.
California State University, Chico G. Karpilovsky
October, 1993
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Contents
Preface vii
Part I Projective Characters 1
1. An Invitation to Projective Characters 3
1.1. Preliminaries 4
1.2. Definitions and elementary properties 7
1.3. Linear independence of 0-characters 15
1.4. Degrees of irreducible projective characters 19
1.5. Projective characters of direct products 28
1.6. Class-function cocycles 32
1.7. Conjugate modules and characters 37
1.8. Mackey’s theorems 40
1.9. Induced projective characters 52
1.10. Brauer’s permutation leiiinia 63
1.11. Orthogonality relations 66
1.11.A. Block ideinpotents and
orthogonality relations 66
l.ll.B. Inner products 70
1.11.C . Generalized orthogonality relations 73
1.11. D. Complex a-characters 78
2. Clifford Theory for Projective Characters 85
2.1. Obstruction cocycles 86
is