Hardcover ISBN:  9780821842713 
Product Code:  FIM/28 
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Electronic ISBN:  9781470417918 
Product Code:  FIM/28.E 
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Book DetailsFields Institute MonographsVolume: 28; 2011; 291 ppMSC: Primary 20; 11; 16;
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 WittGrothendieck 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 BrauerWall group in describing the principal invariants of orthogonal representations and their interplay with each other.ReadershipGraduate students and research mathematicians interested in the representations of finite groups, surgery theory, or equivariant superalgebras.

Table of Contents

Chapters

Chapter 1. Background material

Chapter 2. Isometry representations of finite groups

Chapter 3. Hermitian forms over semisimple algebras

Chapter 4. Equivariant WittGrothendieck and Witt groups

Chapter 5. Representations over finite, local and global fields

Chapter 6. Fröhlich’s invariant, Clifford algebras and the equivariant BrauerWall group


Additional Material

Reviews

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


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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 WittGrothendieck 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 BrauerWall group in describing the principal invariants of orthogonal representations and their interplay with each other.
Graduate students and research mathematicians interested in the representations of finite groups, surgery theory, or equivariant superalgebras.

Chapters

Chapter 1. Background material

Chapter 2. Isometry representations of finite groups

Chapter 3. Hermitian forms over semisimple algebras

Chapter 4. Equivariant WittGrothendieck and Witt groups

Chapter 5. Representations over finite, local and global fields

Chapter 6. Fröhlich’s invariant, Clifford algebras and the equivariant BrauerWall group

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