Pavel Etingof , Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina

Chapter 1. Introduction 1

Chapter 2. Basic notions of representation theory 5
2.1. What is representation theory? 5
2.2. Algebras 8
2.3. Representations 9
2.4. Ideals 15
2.5. Quotients 15
2.6. Algebras defined by generators and relations 16
2.7. Examples of algebras 17
2.8. Quivers 19
2.9. Lie algebras 22
2.10. Historical interlude: Sophus Lie’s trials and transformations 26
2.11. Tensor products 30
2.12. The tensor algebra 35
2.13. Hilbert’s third problem 36
2.14. Tensor products and duals of representations of Lie algebras 36
2.15. Representations of sl(2) 37
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iv Contents
2.16. Problems on Lie algebras 39

Chapter 3. General results of representation theory 41
3.1. Subrepresentations in semisimple representations 41
3.2. The density theorem 43
3.3. Representations of direct sums of matrix algebras 44
3.4. Filtrations 45
3.5. Finite dimensional algebras 46
3.6. Characters of representations 48
3.7. The Jordan-H¨older theorem 50
3.8. The Krull-Schmidt theorem 51
3.9. Problems 53
3.10. Representations of tensor products 56

Chapter 4. Representations of finite groups: Basic results 59
4.1. Maschke’s theorem 59
4.2. Characters 61
4.3. Examples 62
4.4. Duals and tensor products of representations 65
4.5. Orthogonality of characters 65
4.6. Unitary representations. Another proof of Maschke’s theorem for complex representations 68
4.7. Orthogonality of matrix elements 70
4.8. Character tables, examples 71
4.9. Computing tensor product multiplicities using character tables 74
4.10. Frobenius determinant 75
4.11. Historical interlude: Georg Frobenius’s “Principle of Horse Trade” 77
4.12. Problems 81
4.13. Historical interlude: William Rowan Hamilton’s quaternion of geometry, algebra, metaphysics, and poetry 86
Contents v
Chapter 5. Representations of finite groups: Further results 91
5.1. Frobenius-Schur indicator 91
5.2. Algebraic numbers and algebraic integers 93
5.3. Frobenius divisibility 96
5.4. Burnside’s theorem 98
5.5. Historical interlude: William Burnside and intellectual harmony in mathematics 100
5.6. Representations of products 104
5.7. Virtual representations 105
5.8. Induced representations 105
5.9. The Frobenius formula for the character of an induced representation 106
5.10. Frobenius reciprocity 107
5.11. Examples 110
5.12. Representations of Sn 110
5.13. Proof of the classification theorem for representations of Sn 112
5.14. Induced representations for Sn 114
5.15. The Frobenius character formula 115
5.16. Problems 118
5.17. The hook length formula 118
5.18. Schur-Weyl duality for gl(V ) 119
5.19. Schur-Weyl duality for GL(V ) 122
5.20. Historical interlude: Hermann Weyl at the intersection of limitation and freedom 122
5.21. Schur polynomials 128
5.22. The characters of L? 129
5.23. Algebraic representations of GL(V ) 130
5.24. Problems 131
5.25. Representations of GL2(Fq) 132
5.26. Artin’s theorem 141
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5.27. Representations of semidirect products 142

Chapter 6. Quiver representations 145
6.1. Problems 145
6.2. Indecomposable representations of the quivers
A1,A2,A3 150
6.3. Indecomposable representations of the quiver D4 154
6.4. Roots 160
6.5. Gabriel’s theorem 163
6.6. Reflection functors 164
6.7. Coxeter elements 169
6.8. Proof of Gabriel’s theorem 170
6.9. Problems 173

Chapter 7. Introduction to categories 177
7.1. The definition of a category 177
7.2. Functors 179
7.3. Morphisms of functors 181
7.4. Equivalence of categories 182
7.5. Representable functors 183
7.6. Adjoint functors 184
7.7. Abelian categories 186
7.8. Complexes and cohomology 187
7.9. Exact functors 190
7.10. Historical interlude: Eilenberg, Mac Lane, and “general abstract nonsense” 192

Chapter 8. Homological algebra 201
8.1. Projective and injective modules 201
8.2. Tor and Ext functors 203

Chapter 9. Structure of finite dimensional algebras 209
9.1. Lifting of idempotents 209
9.2. Projective covers 210
Contents vii
9.3. The Cartan matrix of a finite dimensional algebra 211
9.4. Homological dimension 212
9.5. Blocks 213
9.6. Finite abelian categories 214
9.7. Morita equivalence 215
References for historical interludes 217
Mathematical references 223
Index 225