A numerical invariant for linear representations of finite groups

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.