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Introduction to orthogonal, symplectic, and unitary representations of finite groups

Author: C R Riehm
Publisher: Providence, R.I. : American Mathematical Society ; Toronto : Fields Institute for Research in Mathematical Sciences, ©2011.
Series: Fields Institute monographs, 28.
Edition/Format:   Print book : EnglishView all editions and formats
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Document Type: Book
All Authors / Contributors: C R Riehm
ISBN: 9780821842713 0821842714
OCLC Number: 725827652
Description: x, 291 p. : ill. ; 26 cm.
Contents: Background material --
Rings and modules ; Simple and semisimple modules and algebras ; Semisimple modules --
Simple and semisimple rings ; Simple and semisimple algebras ; Quaternion algebras --
Brauer group ; Compositums and tensor products of fields --
Involutions on algebras ; Involutions on simple algebras ; Involutions on semisimple algebras --
Formally real fields ; Bilinear and sesquilinear forms ; Definitions and basic properties --
Isometries ; Witt rings ; K real closed or algebraically closed --
Local and global fields --
Trace forms over local fields --
Equivalence of reflexive forms over division algebras --
Grothendieck groups ; Linear representations of finite groups ; Basic facts about linear representations ; Characters twisted contragredient representation --
Induced representations --
Simple components of KG and irreducible representations --
Representations over the real and complex numbers --
Isometry representations of finite groups --
Hermitian forms over semisimple algebras --
Basic properties of -Hermitian forms over semisimple algebras --
Hyperbolic forms --
Witt's Theorem ; algebra of forms ; Sums and products of forms --
Going up ; Going down --
Indecomposable forms --
twisted contragredient representation revisited --
Adjoint involutions ; Forms over simple algebras in terms of matrices --
Transfers of forms over simple algebras --
Morita theory for forms over simple algebras --
Hasse Principle ; Lifting a local representation to a global representation Existence of isometry representations ; Witt's Theorem for isometry representations --
Equivariant Witt-Grothendieck and Witt groups --
Witt-Grothendieck and Witt groups of Hermitian forms over semisimple algebras --
Functorial properties of the Witt-Grothendieck and Witt groups of isometry representations --
graded Witt-Grothendieck and Witt rings --
K0 algebraically closed ; field K0 real closed ; K/K0 definite ; involutory Schur subgroup --
Similarity of isometry representations --
Structure of W (K, G) when K0 is not formally real --
Structure of W(K, G) when K/K0 is a definite extension of algebraic number fields Structure of the Witt group W(K, G) --
Representations over finite, local and global fields --
Symplectic representations ; Unitary and orthogonal representations over finite fields --
Unitary and orthogonal representations over local fields ; Unitary representations ; Orthogonal representations --
Global fields; Isometry representations of the symmetric group --
Frohlich's invariant, Clifford algebras and the equivariant Brauer-Wall group --
Graded algebras and the Brauer-Wall group --
Clifford algebras ; Clifford group ; Equivariant graded algebras ; equivariant Clifford algebra and Witt group --
formula for Frohlich's cohomology class
Series Title: Fields Institute monographs, 28.
Responsibility: Carl R. Riehm.

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