Two remarks on the reduction of Alperin's weight conjecture

TL;DR

This paper demonstrates that a recent reformulation of the inductive McKay condition can be applied to simplify the reduction theorem for Alperin's weight conjecture, especially for simple groups of Lie type.

Contribution

It shows the applicability of Späth's reformulation to Alperin's weight conjecture, streamlining the verification process for certain finite simple groups.

Findings

Reformulation applies to Alperin's weight conjecture reduction theorem

Simplifies checking the inductive condition for Lie type groups

Enhances understanding of the relationship between McKay and Alperin's conjectures

Abstract

The so-called inductive McKay condition on finite simple groups, due to Isaacs-Malle-Navarro (2007), has been recently reformulated by Sp\"ath. We show that this reformulation applies to the reduction theorem for Alperin's weight conjecture, due to Navarro-Tiep (2011). This also simplifies the checking of the inductive condition for Alperin's weight conjecture in the case of simple groups of Lie type with regard to the defining prime.