Character Theory and the McKay ConjectureThe McKay conjecture is the origin of the counting conjectures in the representation theory of finite groups. This book gives a comprehensive introduction to these conjectures, while assuming minimal background knowledge. Character theory is explored in detail along the way, from the very basics to the state of the art. This includes not only older theorems, but some brand new ones too. New, elegant proofs bring the reader up to date on progress in the field, leading to the final proof that if all finite simple groups satisfy the inductive McKay condition, then the McKay conjecture is true. Open questions are presented throughout the book, and each chapter ends with a list of problems, with varying degrees of difficulty. |
Contents
The Basics | 1 |
Action on Characters by Automorphisms | 27 |
Galois Action on Characters | 46 |
Character Values and Identities | 58 |
Characters over a Normal Subgroup | 80 |
Extension of Characters | 104 |
Degrees of Characters | 120 |
The HowlettIsaacs Theorem | 136 |
GlobalLocal Counting Conjectures | 150 |
A Reduction Theorem for the McKay Conjecture | 181 |
Appendix A | 212 |
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Common terms and phrases
abelian action acts affording algebra appeared associated assume automorphisms bijection Brauer’s central Chapter character table character theory character triple characters of G claim classes of G complex conclude condition conjugacy classes conjugate contained Corollary correspondence deduce defined definition determined divides divisible elementary elements exists extends fact field finite group fixes follows Frattini argument function g e G G-invariant given going group G Hence homomorphism hypothesis induction instance integer Irr(G Irr(N irreducible characters irreducible constituent Irrp G Irry Isaacs isomorphic Lemma Let G lies linear McKay conjecture normal notice p-group particular prime Problem projective representation Proof prove Recall representation representatives result satisfies simple groups solvable subgroup subgroup of G Suppose that G Sylow p-subgroup Theorem theory trivial true unique values write zero