A criterion for the inductive Alperin weight condition

TL;DR

This paper introduces a new criterion that simplifies verifying the inductive Alperin weight condition for certain simple groups of Lie type, building on previous criteria for related conjectures in group theory.

Contribution

It provides a novel, simplified criterion for the inductive Alperin weight condition, closely related to existing criteria for the inductive McKay condition, and extends to a criterion for the blockwise version.

Findings

New criterion simplifies checking the inductive Alperin weight condition.

The criterion is closely related to the inductive McKay condition criterion.

Provides a related criterion for the blockwise Alperin weight condition.

Abstract

We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type. It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see [Sp\"a12,2.12]) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro--Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition.