This text covers a variety of topics in representation theory and is intended for graduate students and more advanced researchers who are interested in the field.
The book begins with classical representation theory of finite groups over complex numbers and ends with results on representation theory of quivers. The text includes in particular infinite-dimensional unitary representations for abelian groups, Heisenberg groups and SL(2), and representation theory of finite-dimensional algebras. The last chapter is devoted to some applications of quivers, including Harish-Chandra modules for SL(2). Ample examples are provided and some are revisited with a different approach when new methods are introduced, leading to deeper results. Exercises are spread throughout each chapter.
Prerequisites include an advanced course in linear algebra that covers Jordan normal forms and tensor products as well as basic results on groups and rings.
Author(s): Gruson, Caroline, Serganova, Vera
Year: 2018
Language: English
Pages: 231
Tags: Finite Groups;Quivers ;Representation Theory
Preface......Page 7
Contents......Page 11
1. Definitions and examples......Page 14
2. Ways to produce new representations......Page 16
3. Invariant subspaces and irreducibility......Page 17
4. Characters......Page 20
5. Examples......Page 28
6. Invariant forms......Page 31
7. Representations over mathbbR......Page 34
8. Relationship between representations over mathbbR and over mathbbC......Page 35
1. Modules over associative rings......Page 37
2. Finitely generated modules and Noetherian rings......Page 40
3. The centre of the group algebra k(G)......Page 42
4. One application......Page 45
5. General facts on induced modules......Page 46
6. Induced representations for groups......Page 48
7. Double cosets and restriction to a subgroup......Page 50
8. Mackey's criterion......Page 52
9. Hecke algebras, a first glimpse......Page 53
10. Some examples......Page 54
11. Some general facts about field extension......Page 55
12. Artin's theorem and representations over mathbbQ......Page 57
1. Compact groups......Page 59
2. Orthogonality relations and Peter–Weyl Theorem......Page 67
3. Examples......Page 70
1. Unitary representations of mathbbRn and Fourier transform......Page 76
2. Heisenberg groups and the Stone–von Neumann theorem......Page 81
3. Representations of SL2(mathbbR)......Page 88
2. Semisimple modules and density theorem......Page 92
3. Wedderburn–Artin theorem......Page 95
4. Jordan-Hölder theorem and indecomposable modules......Page 96
5. A bit of homological algebra......Page 101
6. Projective modules......Page 104
7. Representations of Artinian rings......Page 110
8. Abelian categories......Page 114
1. Representations of symmetric groups......Page 116
2. Schur–Weyl duality.......Page 121
3. General facts on Hopf algebras......Page 125
4. The Hopf algebra associated to the representations of symmetric groups......Page 128
5. Classification of PSH algebras part 1: decomposition theorem......Page 130
6. Classification of PSH algebras part 2: unicity for the rank 1 case......Page 132
7. Bases of PSH algebras of rank one......Page 136
8. Harvest......Page 142
9. General linear groups over a finite field......Page 149
1. Representations of quivers......Page 159
2. Path algebra......Page 162
3. Standard resolution and consequences......Page 165
4. Bricks......Page 169
5. Orbits in representation varieties......Page 171
6. Coxeter–Dynkin and affine graphs......Page 173
7. Quivers of finite type and Gabriel's theorem......Page 177
1. Reflection functors......Page 179
3. Weyl group and reflection functors.......Page 182
4. Coxeter functors.......Page 183
5. Further properties of Coxeter functors......Page 184
6. Affine root systems......Page 187
7. Preprojective and preinjective representations......Page 189
8. Regular representations......Page 192
9. Indecomposable representations of affine quivers......Page 199
1. From abelian categories to algebras......Page 202
2. From categories to quivers......Page 204
3. Finitely represented, tame and wild algebras......Page 208
4. Frobenius algebras......Page 209
5. Application to group algebras......Page 211
6. On certain categories of mathfraksl2-modules......Page 214
Bibliography......Page 227
Index......Page 229
