A note on module structures of source algebras of block ideals of finite groups

Let b be a block ideal of the group algebra of a finite group G over an algebraically closed field k of prime characteristic p with a defect group P. Some direct summands, as k[P×P]-module, of a source algebra of the block ideal b outside of the inertial group of a maximal b-Brauer pair will be given; their multiplicities are congruent to 1 modulo p.