Polynomial-Like Behaviour of the Faithful Dimension of p-Groups

Abstract

The faithful dimension of a finite group G over C, denoted by m_faithful(G), is defined to be the smallest integer m such that G can be embedded in GL_m(C). We are interested in computing the faithful dimension of nilpotent p-groups of the form exp(f_{n,c} ⊗_Z R), where f_{n,c} is the free nilpotent Z-Lie algebra of class c on n generators, and R is a finite truncated valuation ring. In the special case of R being a finite field with a sufficiently large characteristic, we obtain a sharp result for the faithful dimension associated with free nilpotent Z-Lie algebras of class c = 4. For a general finite truncated valuation ring R, we obtain asymptotically sharp upper and lower bounds. Our lower bound improves previously known results. Additionally, when R = F_q and c = 5 we compute an upper bound for the faithful dimension of magnitude n^5q^4, and a lower bound of magnitude q^2.