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

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 $k[P\times P]$-modules, 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 modulo p will also be given.

We shall introduce a notion, we shall call it the icc condition, which arises from the isomorphism problem of bimodules over p-subgroups. We shall show for an element $x\in G\setminus N_G\left(P\right)$ which satisfies the $(P,P)$-icc condition, the $k[P\times P]$-module $k\left[PxP\right]$ is isomorphic to a direct summand of the source algebra of b under some further condition. One of our main tools is the Brauer homomorphisms so that the multiplicities will be described using dimensions of the Brauer constructions. Our arguments to investigate these dimensions depend on the Puig's theory, especially multiplicities of points.