Minimal dimension of faithful representations for $p$-groups

TL;DR

This paper determines the minimal dimension of faithful complex representations for certain p-groups, specifically Heisenberg and unitriangular matrix groups over local rings, linking to their essential dimension.

Contribution

It provides exact values for the minimal faithful representation dimension of Heisenberg groups and unitriangular groups over local rings, extending understanding of their representation theory.

Findings

Computed $m_{faithful}(G)$ for Heisenberg groups over $rak{p}^n$

Determined minimal faithful representation dimensions for unitriangular groups

Connected representation dimensions to the groups' essential dimension

Abstract

For a group $G$, we denote by $m_{faithful}(G)$, the smallest dimension of a faithful complex representation of $G$. Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the maximal ideal $\mathfrak{p}$. In this paper, we compute the precise value of $m_{faithful}(G)$ when $G$ is the Heisenberg group over $\mathcal{O}/\mathfrak{p}^n$. We then use the Weil representation to compute the minimal dimension of faithful representations of the group of unitriangular matrices over $\mathcal{O}/\mathfrak{p}^n$ and many of its subgroups. By a theorem of Karpenko and Merkurjev, our result yields the precise value of the essential dimension of the latter finite groups.