Some simple modules for classical groups and $p$-ranks of orthogonal and Hermitian geometries

TL;DR

This paper determines the structure of certain Weyl modules for classical groups, providing insights into their composition factors and submodule lattices, which are crucial for understanding the $p$-ranks of related finite geometries.

Contribution

It explicitly describes the characters and submodule structures of specific Weyl modules for classical groups, advancing the understanding of their representation theory.

Findings

Characters of simple composition factors are determined.

Submodule lattices of Weyl modules are characterized.

Results are applied to compute $p$-ranks of finite geometries.

Abstract

We determine the characters of the simple composition factors and the submodule lattices of certain Weyl modules for classical groups. The results have several applications. The simple modules arise in the study of incidence systems in finite geometries and knowledge of their dimensions yields the $p$-ranks of these incidence systems.