Enumerating traceless matrices over compact discrete valuation rings

TL;DR

This paper counts traceless matrices over certain rings using group statistics, linking algebraic enumeration to combinatorial group theory and representation zeta functions.

Contribution

It introduces a novel enumeration method connecting traceless matrices over valuation rings with Coxeter group statistics and local zeta functions.

Findings

Derived rational generating functions for traceless matrices.

Connected matrix enumeration to symmetric and hyperoctahedral group statistics.

Provided explicit formulas for matrices over finite fields.

Abstract

We enumerate traceless square matrices over finite quotients of compact discrete valuation rings by their image sizes. We express the associated rational generating functions in terms of statistics on symmetric and hyperoctahedral groups, viz. Coxeter groups of types A and B, respectively. These rational functions may also be interpreted as local representation zeta functions associated to the members of an infinite family of finitely generated class-2-nilpotent groups. As a byproduct of our work, we obtain descriptions of the numbers of traceless square matrices over a finite field of fixed rank in terms of statistics on the hyperoctahedral groups.