Polynomiality of the faithful dimension of nilpotent groups over finite truncated valuation rings

Abstract

The faithful dimension of a finite group $\mathrm{G}$ over $\mathbb{C}$, denoted by $m_{\mathrm{faithful}}(\mathrm{G})$, is the smallest integer $n$ such that $\mathrm{G}$ can be embedded in ${\mathrm{GL}}_n(\mathbb{C})$. Continuing our previous work (arXiv:1712.02019), we address the problem of determining the faithful dimension of a finite $p$-group of the form ${\mathcal{G}}_R:=\exp ({\mathfrak{g}}_R)$ associated to ${\mathfrak{g}}_R:=\mathfrak{g}{\otimes }_{\mathbb{Z}}R$ in the Lazard correspondence, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra and $R$ ranges over finite truncated valuation rings. Our first main result is that if $R$ is a finite field with $p^f$ elements and $p$ is sufficiently large, then $m_{\mathrm{faithful}}({\mathcal{G}}_R)=fg(p^f)$ where $g(T)$ belongs to a finite list of polynomials $g_1,\dots ,g_k$, with non-negative integer coefficients. The list of polynomials is uniquely determined by the Lie algebra $\mathfrak{g}$. Furthermore, for $1\le i\le k$ the set of pairs $(p,f)$ for which $g=g_i$ is a finite union of Cartesian products $\mathcal{P}\times \mathcal{F}$, where $\mathcal{P}$ is a Frobenius set of prime numbers and $\mathcal{F}$ is a subset of $\mathbb{N}$ that belongs to the Boolean algebra generated by arithmetic progressions. Next we formulate a conjectural polynomiality property for $m_{\mathrm{faithful}}({\mathcal{G}}_R)$ in the more general setting where $R$ is a finite truncated valuation ring, and prove special cases of this conjecture. In particular, we show that for a vast class of Lie algebras $\mathfrak{g}$ that are defined by partial orders, $m_{\mathrm{faithful}}({\mathcal{G}}_R)$ is given by a single polynomial-type formula. Finally, we compute $m_{\mathrm{faithful}}({\mathcal{G}}_R)$ precisely in the case where $\mathfrak{g}$ is the free metabelian nilpotent Lie algebra of class $c$ on $n$ generators and $R$ is a finite truncated valuation ring.

27 pages, to appear in Transactions of the American Mathematical Society