Low dimensional projective indecomposable modules for Chevalley groups in defining characteristic

TL;DR

This paper establishes new lower bounds for the dimensions of projective indecomposable modules in Chevalley groups, extending previous results and identifying modules of minimal dimension related to Sylow p-subgroups.

Contribution

It generalizes earlier bounds and characterizes modules of minimal dimension, advancing understanding of module structure in Chevalley groups in defining characteristic.

Findings

Modules of dimension equal to the order of Sylow p-subgroups identified

Extended bounds for module dimensions beyond previous results

Provided lower bounds where earlier bounds were vacuous

Abstract

The paper studies lower bounds for the dimensions of projective indecomposable modules for Chevalley groups G in defining characteristic p. The main result extending earlier one by Malle and Weigel (2008) determines the modules in question of dimension equal to the order of a Sylow p-subgroup of G. We also substantially generalize a result by Ballard (1978) on lower bounds for the dimensions of projective indecomposable modules and find lower bounds in some cases where Ballard's bounds are vacuous.