# The Decomposition of Permutation Module for Infinite Chevalley Groups

## TL;DR

This paper fully determines the composition factors of a specific induced module for finite Chevalley groups, advancing understanding of their representation structure over arbitrary fields.

## Contribution

It provides a complete decomposition of the permutation module for infinite Chevalley groups over finite fields, a problem previously unresolved.

## Key findings

- Explicit description of composition factors for the induced module
- Applicable to any field 
- Advances representation theory of Chevalley groups

## Abstract

Let ${\bf G}$ be a connected reductive group defined over $\mathbb{F}_q$, the finite field with $q$ elements. Let ${\bf B}$ be an Borel subgroup defined over $\mathbb{F}_q$. In this paper, we completely determine the composition factors of the induced module $\mathbb{M}(\op{tr})=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}\op{tr}$ ($\op{tr}$ is the trivial ${\bf B}$-module) for any field $\Bbbk$.