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 Lλ

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