Representation growth of the Heisenberg group over $\mathcal{O}[x]/(x^n)$

TL;DR

This paper investigates the representation zeta function of the Heisenberg group over certain polynomial quotient rings, proposing a conjecture and confirming it for small cases, thus advancing understanding of its representation growth.

Contribution

It introduces a conjectured formula for the representation zeta function of the Heisenberg group over $\

Findings

Confirmed the conjecture for n ≤ 3

Proposed a general formula for the zeta function

Raised questions for further research

Abstract

We present a conjectured formula for the representation zeta function of the Heisenberg group over $\mathcal{O}[x]/(x^n)$ where $\mathcal{O}$ is the ring of integers of some number field. We confirm the conjecture for $n\leq 3$ and raise several questions.