Enumerating traceless matrices over compact discrete valuation rings
Group Theory
2017-11-09 v2 Combinatorics
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.
Cite
@article{arxiv.1709.02717,
title = {Enumerating traceless matrices over compact discrete valuation rings},
author = {Angela Carnevale and Shai Shechter and Christopher Voll},
journal= {arXiv preprint arXiv:1709.02717},
year = {2017}
}
Comments
20 pages. Minor revisions, to appear in Israel J. Math