Variants on a conjecture relating block source algebras to characteristic bisets

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.