Mohammad Bardestani, Keivan Mallahi-Karai, Hadi Salmasian
February 9, 2016

### Abstract

For a group G , we denote by mfaithful(G)${m_{\textup{faithful}}(G)}$, the smallest dimension of a faithful complex representation of G . Let F be a non-Archimedean local field with ring of integers 𝒪${\mathcal{O}}$ and maximal ideal 𝔭${\mathfrak{p}}$. In this paper, we compute the precise value of mfaithful(G)${m_{\textup{faithful}}(G)}$ when G is the Heisenberg group over 𝒪/𝔭n${\mathcal{O}/\mathfrak{p}^{n}}$. We then use the Weil representation to compute the minimal dimension of faithful representations of the group of unitriangular matrices over 𝒪/𝔭n${\mathcal{O}/\mathfrak{p}^{n}}$ and many of its subgroups. By a theorem of Karpenko and Merkurjev ([7, Theorem 4.1]), our result yields the precise value of the essential dimension of the latter finite groups.