We will present Conjecture 1.6 introduced by Barker and Gelvin in 2020, 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 investigate if 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.

Joint work with Tiberiu Coconeţ.