Variants on a conjecture relating block source algebras to characteristic bisets

TL;DR

This paper investigates the relationship between source algebras of blocks in finite groups and characteristic bisets of their fusion systems, establishing equivalences under certain conditions.

Contribution

It introduces new conditions under which a basis of a source algebra corresponds to a characteristic biset, advancing understanding of block fusion systems.

Findings

Equivalence of multiple reformulations of the basis being units

Conditions under which the basis forms a characteristic biset

Insight into the structure of source algebras and fusion systems

Abstract

Given a block of a finite group, any source algebra has a basis invariant under the multiplicative actions of the defect group. Is such a basis a characteristic biset of the block fusion system? If the basis can be chosen to consist entirely of units, the question is answered in the affirmative. We prove the equivalence of several reformulations of this stronger condition on the source algebra.