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

Given a finite group G, the faithful dimension of G over C, denoted by m(faithful)(G),is the smallest integer n such that G can be embedded in GL(n)(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 G(R) := exp(g(R)) associated to g(R) := g circle times(Z) R in the Lazard correspondence, where g is a nilpotent 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 mfaithful(G(R)) = fg(p(f)) where g(T) belongs to a finite list of polynomials g1, ... , gk, with non-negative integer coefficients. The latter list of polynomials is uniquely determined by the Lie algebra g. Furthermore, for each 1 <= i <= k the set of pairs (p, f) for which g = g(i) is a finite union of Cartesian products P x F, where P is a Frobenius set of prime numbers and F is a subset of N that belongs to the Boolean algebra generated by arithmetic progressions. Previously, existence of such a polynomial-type formula for mfaithful(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 mfaithful(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 g that are defined by partial orders, mfaithful(G(R)) is given by a single polynomial-type formula.Finally, we compute mfaithful(G(R)) precisely in the case where g is the free metabelian nilpotent Lie algebra of class c on n generators and R is a finite truncated valuation ring.