Group Representation Theory

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, Broue, 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: Broue (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); Thevenaz (The classification of endo-permutaion modules); and Webb (Representations and cohomology of categories).

PrefaceRepresentations, Functors and CohomologyCohomology and Representation TheoryJon F. Carlson1. Introduction2. Modules over p-groups3. Group cohomology4. Support varieties5. The cohomology ring of a dihedral group6. Elementary abelian subgroups in cohomology and representations7. Quillen’s dimension theorem8. Properties of support varieties9. The rank of the group of endotrivial modulesIntroduction to Block TheoryRadha Kessar1. Introduction2. Brauer pairs3. b–Brauer pairs4. Some structure theory5. Alperin’s weight conjecture6. Blocks in characteristic7. Examples of fusion systemsIntroduction to Fusion SystemsMarkus Linckelmann1. Local structure of finite groups2. Fusion systems3. Normalisers and centralisers4. Centric subgroups5. Alperin’s fusion theorem6. Quotients of fusion systems7. Normal fusion systems8. Simple fusion systems9. Normal subsystems and control of fusionEndo-permutation Modules, a Guided TourJacques Th´evenaz1. Introduction2. Endo-permutation modules3. The Dade group4. Examples5. The abelian case6. Some small groups7. Detection of endo-trivial modules8. Classification of endo-trivial modules9. Detection of endo-permutation modules10. Functorial approach11. The dual Burnside ring12. Rational representations and an induction theorem13. Classification of endo-permutation modules14. Consequences of the classificationAn Introduction to the Representations and Cohomology of CategoriesPeter Webb1. Introduction2. The category algebra and some preliminaries3. Restriction and induction of representations4. Parametrization of simple and projective representations5. The constant functor and limits6. Augmentation ideals, derivations and H17. Extensions of categories and H2Algebraic Groups and Finite Reductive Groups An Algebraic Introduction to Complex Reflection GroupsMichel Brou´ePart I. Commutative Algebra: a Crash Course1. Notations, conventions, and prerequisites2. Graded algebras and modules3. Filtrations: associated graded algebras, completion4. Finite ring extensions5. Local or graded k–rings6. Free resolutions and homological dimension7. Regular sequences, Koszul complex, depthPart II. Reflection Groups8. Reflections and roots9. Finite group actions on regular rings10. Ramification and reflecting pairs11. Characterization of reflection groups12. Generalized characteristic degrees and Steinberg theorem13. On the co-invariant algebra14. Isotypic components of the symmetric algebra15. Differential operators, harmonic polynomials16. Orlik-Solomon theorem and first applications17. EigenspacesRepresentations of Algebraic GroupsStephen Donkin1. Algebraic groups and representations2. Representations of semisimple groups3. Truncation to a Levi subgroupModular Representations of Hecke AlgebrasMeinolf Geck1. Introduction2. Harish–Chandra series and Hecke algebras3. Unipotent blocks4. Generic Iwahori–Hecke algebras and specializations5. The Kazhdan–Lusztig basis and the a–function6. Canonical basic sets and Lusztig’s ring J7. The Fock space and canonical bases8. The theorems of Ariki and JaconTopics in the Theory of Algebraic GroupsGary M. Seitz1. Introduction2. Algebraic groups: introduction3. Morphisms of algebraic groups4. Maximal subgroups of classical algebraic groups5. Maximal subgroups of exceptional algebraic groups6. On the finiteness of double coset spaces7. Unipotent elements in classical groups8. Unipotent classes in exceptional groupsBounds for the Orders of the Finite Subgroups of G(k)Jean-Pierre SerreLecture I. History: Minkowski, Schur1. Minkowski2. Schur3. Blichfeldt and othersLecture II. Upper Bounds4. The invariants t and m5. The S-bound6. The M-boundLecture III. Construction of large subgroups7. Statements8. Arithmetic methods (k = Q)9. Proof of theorem 9 for classical groups10. Galois twists11. A general construction12. Proof of theorem 9 for exceptional groups13. Proof of theorems 10 and 1114. The case m = 1Index