Oppositeness in buildings and simple modules for finite groups of Lie type

TL;DR

This paper explores the incidence relations of oppositeness in buildings associated with finite groups of Lie type, linking geometric structures to simple modules in representation theory.

Contribution

It identifies the highest weight of the simple modules arising from oppositeness relations and reduces the problem to cases over prime fields using algebraic group representations.

Findings

Determined the highest weight of the simple modules.

Reduced the problem to prime field cases.

Illustrated methods with specific examples.

Abstract

In the building of a finite group of Lie type we consider the incidence relations defined by oppositeness of flags. Such a relation gives rise to a homomorphism of permutation modules (in the defining characteristic) whose image is a simple module for the group. The $p$-rank of the incidence relation is then the dimension of this simple module. We give some general reductions towards the determination of the character of the simple module. Its highest weight is identified and the problem is reduced to the case of a prime field. The reduced problem can be approached through the representation theory of algebraic groups and the methods are illustrated for some examples.