Minimal dimension of faithful representations for $p$-groups

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.