Jordan types of triangular matrices over a finite field

Abstract

Let $\lambda$ be a partition of an integer $n$ and ${\mathbb{F}}_q$ be a finite field of order $q$. Let $P_{\lambda }(q)$ be the number of strictly upper triangular $n\times n$ matrices of the Jordan type $\lambda$. It is known that the polynomial $P_{\lambda }$ has a tendency to be divisible by high powers of $q$ and $Q=q-1$, and we put $P_{\lambda }(q)=q^{d(\lambda )}{Q}^{e(\lambda )}{R}_{\lambda }(q)$, where $R_{\lambda }(0)\ne 0$ and $R_{\lambda }(1)\ne 0$. In this article, we study the polynomials $P_{\lambda }(q)$ and $R_{\lambda }(q)$. Our main results: an explicit formula for $d(\lambda )$ (an explicit formula for $e(\lambda )$ is known, see Proposition 3.3 below), a recursive formula for $R_{\lambda }(q)$ (a similar formula for $P_{\lambda }(q)$ is known, see Proposition 3.1 below), the stabilization of $R_{\lambda }$ with respect to extending $\lambda$ by adding strings of 1's, and an explicit formula for the limit series $R_{\lambda 1^{\infty }}$. Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.