Counting conjugacy classes in the unipotent radical of parabolic subgroups of $\GL_n(q)$

TL;DR

This paper proves that the number of conjugacy classes in the unipotent radical of certain parabolic subgroups of _q^n is a polynomial in q, providing a new algebraic insight into the structure of these groups.

Contribution

It establishes that for parabolic subgroups stabilizing flags of length at most 5, the conjugacy class count in their unipotent radicals is a polynomial in q.

Findings

Number of conjugacy classes is a polynomial in q

Polynomial has integer coefficients

Valid for flags of length up to 5

Abstract

Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of the general linear group $\GL_n(q)$ that is the stabilizer of a flag in $\FF_q^n$ of length at most 5, and let $U = O_p(P)$. In this note we prove that, as a function of $q$, the number $k(U)$ of conjugacy classes of $U$ is a polynomial in $q$ with integer coefficients.