The faithful dimension of a finite group $G$ is defined as the least dimension of a faithful complex representation of $G$. When $G$ is a finite $p$-group, the faithful dimension of $G$ is also intimately related to the notion of essential dimension, introduced by Buhler and Reichstein.
The problem of determining the faithful dimension for families of $p$-groups arising from ${\mathbb F}_p$-points of a nilpotent algebraic group defined over the field of rational numbers has been studied in some special cases, e.g. the Heisenberg and the full upper-triangular unipotent group. In this talk, I will explain how a variant of Kirillov’s orbit method for finite groups can be used to to address this question for a large family of groups in a uniform fashion. Among other things, it will be shown that the dependence on the prime $p$ is always a piecewise polynomial along Frobenian sets. In specific natural cases, the function turns out to be a single polynomial in $p$. This talk is based on a joint work with M. Bardestani and H. Salmasian.