Reduction theorems for a conjecture on basis in source algebras of blocks of finite groups

TL;DR

This paper presents reduction theorems for a conjecture on the existence of stable unital bases in source algebras of blocks of finite groups, with investigations on blocks of finite simple groups.

Contribution

It provides new reduction theorems for the conjecture and explores the problem in the context of blocks of finite simple groups.

Findings

Reduction theorems for stable unital bases

Verification for blocks of some finite simple groups

Insights into the structure of source algebras

Abstract

The aim of this short research note is to present some results about a conjecture of Barker and Gelvin claiming that any source algebra of a block of a finite group has the unit group containing a basis stabilised by the left and right actions of the defect group. We obtain some reduction theorems for the existence of stable unital basis in source algebras of block algebras. Along the way we investigate this problem for the blocks of some finite simple groups.