  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 essential dimension of
representations can behave in rather unexpected ways in the modular setting.
