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 $R_\lambda(0)\neq0$ and $R_\lambda(1)\neq0$. 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 $R_{\lambda1^\infty}$. Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.