Stable Unital Bases, Hyperfocal Subalgebras and Basic Morita Equivalences

We investigate Conjecture 1.5 introduced by Barker and Gelvin (J. Gr. Theory 25, 973–995 2022), which says that any source algebra of a p-block (p is a prime) of a finite group has the unit group containing a basis stabilized by the left and right actions of the defect group. We will reduce this conjecture to a similar statement about the bases of the hyperfocal subalgebras in the source algebras. We will also show that such unital bases of source algebras of two p-blocks, stabilized by the left and right actions of the defect group, are transported through basic Morita equivalences.