Stable decompositions of certain representations of the finite general linear groups

TL;DR

This paper proves that the irreducible decomposition of certain permutation representations of finite general linear groups stabilizes as the dimension grows, leading to a stability theorem for VIC-modules.

Contribution

It introduces a stabilization result for irreducible decompositions of permutation representations of GL(n,q) and derives a new stability theorem for finitely generated VIC-modules.

Findings

Irreducible decomposition stabilizes for large n

Representation stability for VIC-modules established

Provides new insights into the structure of GL(n,q) representations

Abstract

We prove that the irreducible decomposition of the permutation representation of GL(n,q) on GL(n,q)/GL(n-m,q) stabilizes for large n. We deduce, as a consequence, a representation stability theorem for finitely generated VIC-modules.