The McKay conjecture and Brauer's induction theorem

TL;DR

This paper introduces a refined version of the McKay conjecture, proposing a compatible character correspondence between a finite group and its Sylow normaliser, and verifies it in specific cases, also relating to Broué's conjecture.

Contribution

It proposes a new refinement of the McKay conjecture with compatibility conditions and verifies it in several special cases, advancing understanding of character correspondences.

Findings

Refinement of the McKay conjecture with compatibility conditions.

Verification of the conjecture in specific cases.

Connection to a conjecture of Isaacs and Navarro.

Abstract

Let $G$ be an arbitrary finite group. The McKay conjecture asserts that $G$ and the normaliser $N_G (P)$ of a Sylow $p$-subgroup $P$ in $G$ have the same number of characters of degree not divisible by $p$ (that is, of $p'$-degree). We propose a new refinement of the McKay conjecture, which suggests that one may choose a correspondence between the characters of $p'$-degree of $G$ and $N_G (P)$ to be compatible with induction and restriction in a certain sense. This refinement implies, in particular, a conjecture of Isaacs and Navarro. We also state a corresponding refinement of the Brou\'e abelian defect group conjecture. We verify the proposed conjectures in several special cases.