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

TL;DR

This paper introduces the source permutation module, a block component of the Sylow permutation module in finite group algebras, which remains invariant under splendid Morita equivalences and connects to key conjectures in representation theory.

Contribution

It defines the source permutation module, proves its invariance under Morita equivalences, and explores its structural properties and specific cases in group algebra blocks.

Findings

Source permutation module is invariant under splendid Morita equivalences.

Structural properties relate to self-injectivity and Alperin's weight conjecture.

Explicit calculations for blocks with cyclic and Klein four defect groups.

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.