Group Representation Theory: 1st Edition (Hardback) book cover

Group Representation Theory

1st Edition

Edited by Jacques Thevenaz, Meinolf Geck, Donna Testerman

EPFL Press

350 pages

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Hardback: 9780849392436
pub: 2007-05-07
$150.00
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Description

After the pioneering work of Brauer in the middle of the 20th century in the area of the representation theory of groups, many entirely new developments have taken place and the field has grown into a very large field of study. This progress, and the remaining open problems (e.g., the conjectures of Alterin, Dade, Broué, James, etc.) have ensured that group representation theory remains a lively area of research. In this book, the leading researchers in the field contribute a chapter in their field of specialty, namely: Broué (Finite reductive groups and spetses); Carlson (Cohomology and representations of finite groups); Geck (Representations of Hecke algebras); Seitz (Topics in algebraic groups); Kessar and Linckelmann (Fusion systems and blocks); Serre (On finite subgroups of Lie groups); Thévenaz (The classification of endo-permutaion modules); and Webb (Representations and cohomology of categories).

Table of Contents

Preface

Representations, Functors and Cohomology

Cohomology and Representation Theory

Jon F. Carlson

1. Introduction

2. Modules over p-groups

3. Group cohomology

4. Support varieties

5. The cohomology ring of a dihedral group

6. Elementary abelian subgroups in cohomology and representations

7. Quillen’s dimension theorem

8. Properties of support varieties

9. The rank of the group of endotrivial modules

Introduction to Block Theory

Radha Kessar

1. Introduction

2. Brauer pairs

3. b–Brauer pairs

4. Some structure theory

5. Alperin’s weight conjecture

6. Blocks in characteristic

7. Examples of fusion systems

Introduction to Fusion Systems

Markus Linckelmann

1. Local structure of finite groups

2. Fusion systems

3. Normalisers and centralisers

4. Centric subgroups

5. Alperin’s fusion theorem

6. Quotients of fusion systems

7. Normal fusion systems

8. Simple fusion systems

9. Normal subsystems and control of fusion

Endo-permutation Modules, a Guided Tour

Jacques Th´evenaz

1. Introduction

2. Endo-permutation modules

3. The Dade group

4. Examples

5. The abelian case

6. Some small groups

7. Detection of endo-trivial modules

8. Classification of endo-trivial modules

9. Detection of endo-permutation modules

10. Functorial approach

11. The dual Burnside ring

12. Rational representations and an induction theorem

13. Classification of endo-permutation modules

14. Consequences of the classification

An Introduction to the Representations and Cohomology of Categories

Peter Webb

1. Introduction

2. The category algebra and some preliminaries

3. Restriction and induction of representations

4. Parametrization of simple and projective representations

5. The constant functor and limits

6. Augmentation ideals, derivations and H1

7. Extensions of categories and H2

Algebraic Groups and Finite Reductive Groups

An Algebraic Introduction to Complex Reflection Groups

Michel Brou´e

Part I. Commutative Algebra: a Crash Course

1. Notations, conventions, and prerequisites

2. Graded algebras and modules

3. Filtrations: associated graded algebras, completion

4. Finite ring extensions

5. Local or graded k–rings

6. Free resolutions and homological dimension

7. Regular sequences, Koszul complex, depth

Part II. Reflection Groups

8. Reflections and roots

9. Finite group actions on regular rings

10. Ramification and reflecting pairs

11. Characterization of reflection groups

12. Generalized characteristic degrees and Steinberg theorem

13. On the co-invariant algebra

14. Isotypic components of the symmetric algebra

15. Differential operators, harmonic polynomials

16. Orlik-Solomon theorem and first applications

17. Eigenspaces

Representations of Algebraic Groups

Stephen Donkin

1. Algebraic groups and representations

2. Representations of semisimple groups

3. Truncation to a Levi subgroup

Modular Representations of Hecke Algebras

Meinolf Geck

1. Introduction

2. Harish–Chandra series and Hecke algebras

3. Unipotent blocks

4. Generic Iwahori–Hecke algebras and specializations

5. The Kazhdan–Lusztig basis and the a–function

6. Canonical basic sets and Lusztig’s ring J

7. The Fock space and canonical bases

8. The theorems of Ariki and Jacon

Topics in the Theory of Algebraic Groups

Gary M. Seitz

1. Introduction

2. Algebraic groups: introduction

3. Morphisms of algebraic groups

4. Maximal subgroups of classical algebraic groups

5. Maximal subgroups of exceptional algebraic groups

6. On the finiteness of double coset spaces

7. Unipotent elements in classical groups

8. Unipotent classes in exceptional groups

Bounds for the Orders of the Finite Subgroups of G(k)

Jean-Pierre Serre

Lecture I. History: Minkowski, Schur

1. Minkowski

2. Schur

3. Blichfeldt and others

Lecture II. Upper Bounds

4. The invariants t and m

5. The S-bound

6. The M-bound

Lecture III. Construction of large subgroups

7. Statements

8. Arithmetic methods (k = Q)

9. Proof of theorem 9 for classical groups

10. Galois twists

11. A general construction

12. Proof of theorem 9 for exceptional groups

13. Proof of theorems 10 and 11

14. The case m = 1

Index

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT022000
MATHEMATICS / Number Theory