The source permutation module of a block of a finite group algebra

Abstract

For $G$ a finite group, $k$ a field of prime characteristic $p$, and $S$ a Sylow $p$-subgroup of $G$, the Sylow permutation module $\mathrm{Ind}^G_S(k)$ plays a role in diverse facets of representation theory and group theory, ranging from Alperin's weight conjecture to statistical considerations of $S$-$S$-double cosets in $G$. The Sylow permutation module breaks up along the block decomposition of the group algebra $kG$, but the resulting block components are not invariant under splendid Morita equivalences. We introduce a summand of the block component, which we call source permutation module, which is shown to be invariant under such equivalences. We investigate general structural properties of the source permutation module and we show that well-known results relating the self-injectivity of the endomorphism algebra of the Sylow permutation to Alperin's weight conjecture carry over to the source permutation module. We calculate this module in various cases, such as certain blocks with cyclic defect group and blocks with a Klein four defect group, and for some blocks of symmetric groups, prompted by a question in a recent paper by Diaconis-Giannelli-Guralnick-Law-Navarro-Sambale-Spink on the self-injectivity of the endomorphism algebra of the Sylow permutation module for symmetric groups.