**Fields Institute Monographs**

Volume: 28;
2011;
291 pp;
Hardcover

MSC: Primary 20; 11; 16;

**Print ISBN: 978-0-8218-4271-3
Product Code: FIM/28**

List Price: $111.00

AMS Member Price: $88.80

MAA Member Price: $99.90

**Electronic ISBN: 978-1-4704-1791-8
Product Code: FIM/28.E**

List Price: $104.00

AMS Member Price: $83.20

MAA Member Price: $93.60

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#### Supplemental Materials

# Introduction to Orthogonal, Symplectic and Unitary Representations of Finite Groups

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*Carl R. Riehm*

A co-publication of the AMS and Fields Institute

Orthogonal, symplectic and unitary
representations of finite groups lie at the crossroads of two more
traditional subjects of mathematics—linear representations of finite
groups, and the theory of quadratic, skew symmetric and Hermitian
forms—and thus inherit some of the characteristics of both.

This book is written as an introduction to the subject and not as
an encyclopaedic reference text. The principal goal is an exposition
of the known results on the equivalence theory, and related matters
such as the Witt and Witt-Grothendieck groups, over the
“classical” fields—algebraically closed, real closed,
finite, local and global. A detailed exposition of the background
material needed is given in the first chapter.

It was A. Fröhlich who first gave a systematic organization of
this subject, in a series of papers beginning in 1969. His paper
Orthogonal and symplectic representations of groups represents the
culmination of his published work on orthogonal and symplectic
representations. The author has included most of the work from that paper,
extending it to include unitary representations, and also providing
new approaches, such as the use of the equivariant Brauer-Wall group
in describing the principal invariants of orthogonal representations
and their interplay with each other.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Readership

Graduate students and research mathematicians interested in the representations of finite groups, surgery theory, or equivariant superalgebras.

#### Reviews & Endorsements

This book is a most welcome introduction to this wonderful subject with deep roots in two classical areas of mathematics, collecting in one place the most recent developments. It can be profitably read by anyone interested with a basic background on representation theory and quadratic forms.

-- MAA Reviews

#### Table of Contents

# Table of Contents

## Introduction to Orthogonal, Symplectic and Unitary Representations of Finite Groups

- Cover Cover11
- Title page i2
- Contents iii4
- Preface vii8
- Notation ix10
- Background material 112
- Isometry representations of finite groups 8394
- Hermitian forms over semisimple algebras 91102
- Equivariant Witt-Grothendieck and Witt groups 151162
- Representations over finite, local and global fields 219230
- Fröhlich’s invariant, Clifford algebras and the equivariant Brauer-Wall group 253264
- Bibliography 279290
- Glossary 283294
- Index 287298
- Back Cover Back Cover1305