The McKay Conjecture on character degrees

TL;DR

This paper proves McKay's conjecture relating the number of irreducible characters of prime-to-ell degree in finite groups to normalizers of Sylow ell-subgroups, completing a reduction to simple groups using classification.

Contribution

It completes the proof of McKay's conjecture by verifying a key reduction step for finite simple groups of type D, using detailed character analysis of specific subgroups.

Findings

Proved McKay's conjecture for all finite groups.

Characterized the irreducible characters of certain reductive subgroups.

Showed the action of automorphisms on these characters.

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.