A numerical invariant for linear representations of finite groups

Abstract

We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized Severi–Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that in the modular setting ed($\rho$) can be arbitrary large (under a mild assumption on $G$). Here $G$ is fixed, and $\rho$ is allowed to range over the finite-dimensional representations of $G$. The appendix gives a constructive version of this result.