Reduction theorems for a conjecture on bases in source algebras of blocks of finite groups

The aim of this short research note is to present some results about a conjecture of Barker and Gelvin [J. Group Theory 25 (2022), pp. 973–995, Conjecture 1.5 ] claiming that any source algebra of a p p -block ( p 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 obtain some reduction theorems for the existence of stable unital basis in source algebras of p p -block algebras. Along the way we investigate this problem for the p p -blocks of some finite simple groups.