Kirillov's orbit method and polynomiality of the faithful dimension of $p$-groups

Abstract

Given a finite group $\mathrm{G}$ and a field $K$, the faithful dimension of $\mathrm{G}$ over $K$ is defined to be the smallest integer $n$ such that $\mathrm{G}$ embeds into ${\mathrm{GL}}_n(K)$. In this paper we address the problem of determining the faithful dimension of a $p$-group of the form ${\mathcal{G}}_q:=\exp (\mathfrak{g}{\otimes }_{\mathbb{Z}}{\mathbb{F}}_q)$ associated to ${\mathfrak{g}}_q:=\mathfrak{g}{\otimes }_{\mathbb{Z}}{\mathbb{F}}_q$ in the Lazard correspondence, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra which is finitely generated as an abelian group. We show that in general the faithful dimension of ${\mathcal{G}}_p$ is a piecewise polynomial function of $p$ on a partition of primes into Frobenius sets. Furthermore, we prove that for $p$ sufficiently large, there exists a partition of $\mathbb{N}$ by sets from the Boolean algebra generated by arithmetic progressions, such on each part the faithful dimension of ${\mathcal{G}}_q$ for $q:=p^f$ is equal to $fg(p^f)$ for a polynomial $g(T)$. We show that for many naturally arising $p$-groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single formula of the latter form. The arguments rely on various tools from number theory, model theory, combinatorics and Lie theory.