Jordan Types of Triangular Matrices over a Finite Field

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.