Quasi-isolated blocks and the Alperin-McKay conjecture

TL;DR

This paper advances the proof of the Alperin-McKay conjecture by reducing the verification of the inductive condition to isolated blocks, leading to a proof for 2-blocks with abelian defect.

Contribution

It shows that verifying the inductive Alperin-McKay condition can be simplified to isolated blocks, improving the approach to the conjecture.

Findings

Reduction of the verification process to isolated blocks

Proof of the conjecture for 2-blocks with abelian defect

Simplification of the inductive condition verification

Abstract

The Alperin-McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Sp\"ath showed that the Alperin-McKay conjecture holds if the so-called inductive Alperin-McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this we obtain a proof of the Alperin-McKay conjecture for 2-blocks of finite groups with abelian defect.