Representations and Cohomology: Volume 1, Basic Representation Theory of Finite Groups and Associative AlgebrasNow in paperback this is the second of two volumes that will provide an introdcution to modern developments in the representation theory of finite groups and associative algebras. The subject is viewed from the perspective of homological algebra and the theory of representations of finite dimensional algebras; the author emphasises modular representations and the homological algebra associated with their categories.This volume concentrates on the cohomology of groups, always with representations in view. |
Contents
Background material from rings and modules | 1 |
12 The Jacobson radical | 3 |
13 The Wedderburn structure theorem | 5 |
14 The KrullSchmidt theorem | 7 |
15 Projective and injective modules | 8 |
16 Frobenius and symmetric algebras | 11 |
17 Idempotents and the Cartan matrix | 12 |
18 Blocks and central idempotents | 15 |
44 Representation type of algebras | 114 |
45 Dynkin and Euclidean Diagrams | 116 |
46 Weyl groups and Coxeter transformations | 120 |
47 Path algebras of finite type | 124 |
48 Functor categories | 128 |
49 The Auslander algebra | 130 |
410 Functorial nitrations | 132 |
411 Representations of dihedral groups | 135 |
19 Algebras over a complete domain | 17 |
Homological algebra | 21 |
22 Morita theory | 25 |
23 Chain complexes and homology | 27 |
24 Ext and Tor | 30 |
25 Long exact sequences | 35 |
26 Extensions | 38 |
27 Operations on chain complexes | 42 |
28 Induction and restriction | 46 |
Modules for group algebras | 49 |
32 Cup products | 56 |
33 Induction and restriction | 60 |
34 Standard resolutions | 63 |
35 Cyclic and abelian groups | 65 |
36 Relative projectivity and transfer | 68 |
37 Low degree cohomology | 75 |
38 Stable elements | 79 |
39 Relative cohomology | 81 |
310 Vertices and sources | 83 |
311 Trivial source modules | 84 |
312 Green correspondence | 85 |
313 Clifford theory | 87 |
314 Modules for pgroups | 91 |
315 Tensor induction | 96 |
Methods from the representations of algebras | 99 |
42 Finite dimensional hereditary algebras | 106 |
43 Representations of the Klein four group | 107 |
412 Almost split sequences | 143 |
413 Irreducible maps and the AuslanderReiten quiver | 149 |
414 Rojters theorem | 153 |
415 The Riedtmann structure theorem | 154 |
416 Tubes | 158 |
417 Webbs Theorem | 159 |
418 Brauer graph algebras | 165 |
Representation rings and Burnside rings | 171 |
51 Representation rings and Grothendieck rings | 172 |
52 Ordinary character theory | 174 |
53 Brauer character theory | 175 |
54 Gsets and the Burnside ring | 177 |
55 The trivial source ring | 183 |
56 Induction theorems | 185 |
57 Relatively projective and relatively split ideals | 190 |
58 A quotient without nilpotent elements | 191 |
59 Psi operations | 193 |
510 Bilinear forms on representation rings | 196 |
511 Nonsingularity | 198 |
Block theory | 201 |
62 The Brauer map | 204 |
63 Brauers second main theorem | 207 |
64 Clifford theory of blocks | 208 |
65 Blocks of cyclic defect | 212 |
66 Klein four defect groups | 218 |
Bibliography | 225 |
| 237 | |
Common terms and phrases
Auslander-Reiten quiver automorphism block of kG Brauer graph chain complexes cohomology commutative composition factors conjugate COROLLARY coset cyclic decomposition defect group define DEFINITION denote dimension direct sum direct summand Dynkin diagram Euclidean diagram ExtRG field of characteristic finite dimensional algebra finite group finite representation type finitely generated A-module follows functor given Green correspondence group algebra hence HomRG HomɅ idempotent indecomposable modules induction theorem infinite injective inverse kernel Krull-Schmidt theorem left A-module Lemma M₁ M₂ Math morphism multiplicity non-zero p-group p-subgroup path algebra projective cover projective indecomposable projective modules projective relative projective resolution PROOF Proposition quotient representation ring representation theory RG-module ring homomorphism Section short exact sequence simple modules split sequences subadditive function subgroup of G submodule Suppose surjective trivial source U₁ unique uniserial V₁ vector space vertex vertices write zero