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The McKay Conjecture on character degrees

Marc Cabanes, Britta Späth

Abstract

We prove that for any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups. This equality was conjectured by John McKay. The conjecture was reduced by Isaacs--Malle--Navarro (2007) to a conjecture on representations, linear and projective, of finite simple groups that we finish proving here using the classification of those groups. We study mainly characters of normalizers N$_{\mathbf G}({\mathbf S})^F$ of Sylow $d$-tori ${\mathbf S}$ ($d\geq 3$) in a simply-connected algebraic group ${\mathbf G}$ of type D$_l$ ($l\geq 4$) for which $F$ is a Frobenius endomorphism. We also introduce a certain class of $F$-stable reductive subgroups ${\mathbf M}\leq {\mathbf G}$ of maximal rank where ${\mathbf M}^\circ$ is of type some D$_{k}\times\ $D$_{l-k}$. The finite groups ${\mathbf M}^F$ are an efficient substitute for N$_{\mathbf G}({\mathbf S})^F$ or the $\ell$-local subgroups of ${\mathbf G}^F$ relevant to McKay's abstract statement. For a general class of those subgroups ${\mathbf M}^F$ we describe their characters and the action of Aut$({\mathbf G}^F)_{{\mathbf M}^F}$ on them, showing in particular that Irr$({\mathbf M}^F)$ and Irr$({\mathbf G}^F)$ share some key features in that regard.

The McKay Conjecture on character degrees

Abstract

We prove that for any prime , any finite group has as many irreducible complex characters of degree prime to as the normalizers of its Sylow -subgroups. This equality was conjectured by John McKay. The conjecture was reduced by Isaacs--Malle--Navarro (2007) to a conjecture on representations, linear and projective, of finite simple groups that we finish proving here using the classification of those groups. We study mainly characters of normalizers N of Sylow -tori () in a simply-connected algebraic group of type D () for which is a Frobenius endomorphism. We also introduce a certain class of -stable reductive subgroups of maximal rank where is of type some DD. The finite groups are an efficient substitute for N or the -local subgroups of relevant to McKay's abstract statement. For a general class of those subgroups we describe their characters and the action of Aut on them, showing in particular that Irr and Irr share some key features in that regard.

Paper Structure

This paper contains 37 sections, 67 theorems, 162 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The McKay conjecture
  3. Early results and perspectives
  4. A conjecture made (quasi) simple
  5. Quasisimple groups of Lie type
  6. Proving (iMK) for the non-defining primes
  7. The present paper
  8. Background results
  9. Characters and extensions
  10. Character triples and inductive McKay condition
  11. Groups of Lie type, generators and automorphisms
  12. The conditions (iMK), $\mathbf{A}(d)$ and $\mathbf{B}(d)$
  13. The curious case of types D and B
  14. Centralizers of semisimple elements and consequences for $\mathrm{Irr}({{{{\mathbf G}}^F}})$
  15. Centralizers of semisimple elements
  16. ...and 22 more sections

Key Result

Theorem A

Let $X$ be a finite group, $\ell$ a prime and $S$ a Sylow $\ell$-subgroup of $X$. Let $\mathrm{Irr}_{\ell ' }(X)$ denote the set of complex irreducible characters of $X$ whose degree is prime to $\ell$. Then

Theorems & Definitions (140)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 130 more