On the vertices of indecomposable summands of certain Lefschetz modules

TL;DR

This paper investigates the structure of indecomposable summands of Lefschetz modules related to p-radical and p-centric subgroups in groups with parabolic characteristic p, revealing their connections to Lie type groups and block theory.

Contribution

It characterizes non-projective indecomposable summands of Lefschetz modules as Green correspondents of inflated Steinberg modules in groups with certain Lie type components.

Findings

Non-projective summands lie in non-principal blocks.

Summands are Green correspondents of inflated Steinberg modules.

Vertices of summands are defect groups of the blocks.

Abstract

We study the reduced Lefschetz module of the complex of p-radical and p-centric subgroups. We assume that the underlying group G has parabolic characteristic p and the centralizer of a certain noncentral p-element has a component with central quotient H a finite group of Lie type in characteristic p. A non-projective indecomposable summand of the associated Lefschetz module lies in a non-principal block of G and it is a Green correspondent of an inflated, extended Steinberg module for a Lie subgroup of H. The vertex of this summand is the defect group of the block in which it lies.