Group Representation Theory

Abstract

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