Mackey decomposition for Brauer pairs

For a finite group G and an algebraically closed field k of characteristic p, a k-algebra A with a G-action is called a G-algebra. A pair (P,c) such that P is a p-subgroup of G and c is a block idempotent of the G-algebra A(P)is called a Brauer pair. Brauer pairs form a refinement of the G-poset of p-subgroups of a finite group G. We define the ordinary Mackey category B of Brauer pairs on an interior p-permutation G-algebra A over an algebraically closed field k of characteristic p. We then show that, given a field K of characteristic zero and a primitive idempotent f ∈ AG, then the category algebra of Bf over K is semisimple.