Stable unital basis, hyperfocal subalgebras and basic Morita equivalences

TL;DR

This paper explores the structure of source algebras of p-blocks in finite groups, focusing on unital bases stabilized by defect groups, and demonstrates their invariance under basic Morita equivalences.

Contribution

It reduces a conjecture about basis stabilization in source algebras to hyperfocal subalgebras and shows these bases are preserved under basic Morita equivalences.

Findings

Reduction of the conjecture to hyperfocal subalgebras

Establishment of basis invariance under Morita equivalences

Insights into the structure of source algebras in modular representation theory

Abstract

We investigate Conjecture 1.6 introduced by Barker and Gelvin in [3], which says that any source algebra of a p-block (p is a prime) of finite group has the unit group containing a basis stabilized by left and right action of the defect group. We will reduce this conjecture to a similar statement about basis of the hyperfocal subalgebras in the source algebra. We will also show that such unital basis of source algebras of two p-blocks, stabilized by left and right action of the defect group, are transported trough basic Morita equivalences.