A note on module structures of source algebras of block ideals of finite groups. II (Q6668559)

Let \(G\) be a finite group, \(p \in \pi(G)\), \(b\) a block idempotent of the group algebra \(kG\) over an algebraically closed field \(k\) of characteristic \(p\) with a defect group \(P\) (see the previous paper by \textit{T. Okuyama} and the author, [J. Algebra 497, 92--101 (2018; Zbl 1477.20023)]).\N\NIn the paper under review, the author describes some direct summands, viewed as \(k[P\times P]\)-modules, of a source algebra of the block ideal \(b\) outside the inertial group of a maximal \(b\)-Brauer pair, along with their multiplicities modulo \(p\). A concept referred to as the \textsf{icc} condition is introduced, which arises from the isomorphism problem of bimodules over \(p\)-subgroups. It is shown that for an element \(x\in G \setminus N_{G}(P)\) satisfying the \((P,P)\)-\textsf{icc} condition, the \(k[P\times P]\)-module \(k[PxP]\) is isomorphic to a direct summand of the source algebra of \(b\), under certain additional conditions. One of the main tools employed is the Brauer homomorphism, which is used to describe these multiplicities in terms of the dimensions of Brauer constructions. The investigation of these dimensions relies heavily on Puig's theory, particularly on the study of multiplicities of points.