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