Pavel Etingof , Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina
Chapter 1. Introduction 1Chapter 2. Basic notions of representation theory 52.1. What is representation theory? 52.2. Algebras 82.3. Representations 92.4. Ideals 152.5. Quotients 152.6. Algebras defined by generators and relations 162.7. Examples of algebras 172.8. Quivers 192.9. Lie algebras 222.10. Historical interlude: Sophus Lie’s trials and transformations 262.11. Tensor products 302.12. The tensor algebra 352.13. Hilbert’s third problem 362.14. Tensor products and duals of representations of Lie algebras 362.15. Representations of sl(2) 37iiiiv Contents2.16. Problems on Lie algebras 39Chapter 3. General results of representation theory 413.1. Subrepresentations in semisimple representations 413.2. The density theorem 433.3. Representations of direct sums of matrix algebras 443.4. Filtrations 453.5. Finite dimensional algebras 463.6. Characters of representations 483.7. The Jordan-H¨older theorem 503.8. The Krull-Schmidt theorem 513.9. Problems 533.10. Representations of tensor products 56Chapter 4. Representations of finite groups: Basic results 594.1. Maschke’s theorem 594.2. Characters 614.3. Examples 624.4. Duals and tensor products of representations 654.5. Orthogonality of characters 654.6. Unitary representations. Another proof of Maschke’s theorem for complex representations 684.7. Orthogonality of matrix elements 704.8. Character tables, examples 714.9. Computing tensor product multiplicities using character tables 744.10. Frobenius determinant 754.11. Historical interlude: Georg Frobenius’s “Principle of Horse Trade” 774.12. Problems 814.13. Historical interlude: William Rowan Hamilton’s quaternion of geometry, algebra, metaphysics, and poetry 86Contents vChapter 5. Representations of finite groups: Further results 915.1. Frobenius-Schur indicator 915.2. Algebraic numbers and algebraic integers 935.3. Frobenius divisibility 965.4. Burnside’s theorem 985.5. Historical interlude: William Burnside and intellectual harmony in mathematics 1005.6. Representations of products 1045.7. Virtual representations 1055.8. Induced representations 1055.9. The Frobenius formula for the character of an induced representation 1065.10. Frobenius reciprocity 1075.11. Examples 1105.12. Representations of Sn 1105.13. Proof of the classification theorem for representations of Sn 1125.14. Induced representations for Sn 1145.15. The Frobenius character formula 1155.16. Problems 1185.17. The hook length formula 1185.18. Schur-Weyl duality for gl(V ) 1195.19. Schur-Weyl duality for GL(V ) 1225.20. Historical interlude: Hermann Weyl at the intersection of limitation and freedom 1225.21. Schur polynomials 1285.22. The characters of L? 1295.23. Algebraic representations of GL(V ) 1305.24. Problems 1315.25. Representations of GL2(Fq) 1325.26. Artin’s theorem 141vi Contents5.27. Representations of semidirect products 142Chapter 6. Quiver representations 1456.1. Problems 1456.2. Indecomposable representations of the quiversA1,A2,A3 1506.3. Indecomposable representations of the quiver D4 1546.4. Roots 1606.5. Gabriel’s theorem 1636.6. Reflection functors 1646.7. Coxeter elements 1696.8. Proof of Gabriel’s theorem 1706.9. Problems 173Chapter 7. Introduction to categories 1777.1. The definition of a category 1777.2. Functors 1797.3. Morphisms of functors 1817.4. Equivalence of categories 1827.5. Representable functors 1837.6. Adjoint functors 1847.7. Abelian categories 1867.8. Complexes and cohomology 1877.9. Exact functors 1907.10. Historical interlude: Eilenberg, Mac Lane, and “general abstract nonsense” 192Chapter 8. Homological algebra 2018.1. Projective and injective modules 2018.2. Tor and Ext functors 203Chapter 9. Structure of finite dimensional algebras 2099.1. Lifting of idempotents 2099.2. Projective covers 210Contents vii9.3. The Cartan matrix of a finite dimensional algebra 2119.4. Homological dimension 2129.5. Blocks 2139.6. Finite abelian categories 2149.7. Morita equivalence 215References for historical interludes 217Mathematical references 223Index 225