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

TL;DR

This paper investigates the faithful dimension of certain finite $p$-groups derived from nilpotent Lie algebras over finite rings, establishing polynomial formulas and conjectures for their minimal faithful representations.

Contribution

It provides explicit polynomial formulas for the faithful dimension over finite fields and formulates a conjecture for more general rings, with partial proofs and exact computations for specific Lie algebras.

Findings

Faithful dimension is given by polynomial functions for large primes.

The set of parameters where each polynomial applies is described by Frobenius sets and arithmetic progressions.

Exact faithful dimension computed for free metabelian nilpotent Lie algebras.

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 $g_1,\ldots,g_k$, with non-negative integer coefficients. The list of polynomials is uniquely determined by the Lie algebra $\mathfrak g$. Furthermore, for $1\leq i\leq k$ 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.