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

Abstract:

Given a finite group $\mathrm {G}$, the faithful dimension of $\mathrm {G}$ 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 the work initiated by Bardestani et al. [Compos. Math. 155 (2019), pp. 1618–1654], we address the problem of determining the faithful dimension of a finite $p$-group of the form $\mathscr {G}_R≔\exp (\mathfrak {g}_R)$ 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 $m_\mathrm {faithful}(\mathscr {G}_R)=fg(p^f)$ where $g(T)$ belongs to a finite list of polynomials $g_1,\ldots ,g_k$, with non-negative integer coefficients. The latter list of polynomials is uniquely determined by the Lie algebra $\mathfrak {g}$. Furthermore, for each $1\le i\leq k$ the set of pairs $(p,f)$ for which $g=g_i$ is a finite union of Cartesian products $\mathscr P\times \mathscr F$, where $\mathscr P$ is a Frobenius set of prime numbers and $\mathscr F$ is a subset of $\mathbb N$ that belongs to the Boolean algebra generated by arithmetic progressions. Previously, existence of such a polynomial-type formula for $m_\mathrm {faithful}(\mathscr {G}_R)$ was only established under the assumption that either $f=1$ or $p$ is fixed.

Next we formulate a conjectural polynomiality property for the value of $m_\mathrm {faithful}(\mathscr {G}_R)$ 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, $m_\mathrm {faithful}(\mathscr {G}_R)$ is given by a single polynomial-type formula.

Finally, we compute $m_\mathrm {faithful}(\mathscr {G}_R)$ 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.