$U_n(q)$ acting on flags and supercharacters

TL;DR

This paper studies how the group of lower unitriangular matrices acts on flags in a vector space and decomposes related permutation representations into irreducible modules, advancing understanding of supercharacters and subgroup actions.

Contribution

It provides a detailed analysis of the restriction of $GL_n(q)$ actions to $U_n(q)$ on flags and achieves a complete decomposition for flags of length two.

Findings

Complete decomposition of permutation representation on cosets of maximal parabolic subgroups.

Explicit description of $U_n(q)$-module structure on flag actions.

Enhanced understanding of supercharacters for $U_n(q)$.

Abstract

Let $U=U_n(q)$ be the group of lower unitriangular $n \times n$-matrices with entries in the field $\mathbb F_q$ with $q$ elements for some prime power $q$ and $n \in \mathbb N$. We investigate the restriction to $U$ of the permutation action of $GL_n(q)$ on flags in the natural $GL_n(q)$-module $\mathbb F_q^n$. Applying our results to the special case of flags of length two we obtain a complete decomposition of the permutation representation of $GL_n(q)$ on the cosets of maximal parabolic subgroups into irreducible $\mathbb C U$-modules.