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

Abstract

Given a finite group G \mathrm {G} , the faithful dimension of G \mathrm {G} over C \mathbb {C} , denoted by m f a i t h f u l ( G ) m_\mathrm {faithful}(\mathrm {G}) , is the smallest integer n n such that G \mathrm {G} can be embedded in G L n ( C ) \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 p -group of the form G R ≔ exp ⁡ ( g R ) \mathscr {G}_R≔\exp (\mathfrak {g}_R) associated to g R ≔ g ⊗ Z R \mathfrak {g}_R≔\mathfrak {g}\otimes _\mathbb {Z}R in the Lazard correspondence, where g \mathfrak {g} is a nilpotent Z \mathbb {Z} -Lie algebra and R R ranges over finite truncated valuation rings. Our first main result is that if R R is a finite field with p f p^f elements and p p is sufficiently large, then m f a i t h f u l ( G R ) = f g ( p f ) m_\mathrm {faithful}(\mathscr {G}_R)=fg(p^f) where g ( T ) g(T) belongs to a finite list of polynomials g 1 , … , g k g_1,\ldots ,g_k , with non-negative integer coefficients. The latter list of polynomials is uniquely determined by the Lie algebra g \mathfrak {g} . Furthermore, for each 1 ≤ i ≤ k 1\le i\leq k the set of pairs ( p , f ) (p,f) for which g = g i g=g_i is a finite union of Cartesian products P × F \mathscr P\times \mathscr F , where P \mathscr P is a Frobenius set of prime numbers and F \mathscr F is a subset of N \mathbb N that belongs to the Boolean algebra generated by arithmetic progressions. Previously, existence of such a polynomial-type formula for m f a i t h f u l ( G R ) m_\mathrm {faithful}(\mathscr {G}_R) was only established under the assumption that either f = 1 f=1 or p p is fixed. Next we formulate a conjectural polynomiality property for the value of m f a i t h f u l ( G R ) m_\mathrm {faithful}(\mathscr {G}_R) in the more general setting where R 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 \mathfrak {g} that are defined by partial orders, m f a i t h f u l ( G R ) m_\mathrm {faithful}(\mathscr {G}_R) is given by a single polynomial-type formula. Finally, we compute m f a i t h f u l ( G R ) m_\mathrm {faithful}(\mathscr {G}_R) precisely in the case where g \mathfrak {g} is the free metabelian nilpotent Lie algebra of class c c on n n generators and R R is a finite truncated valuation ring.