Compression of finite group actions and covariant dimension

Let G be a finite group and f: V -< W an equivariant morphism of finite dimensional G-modules, classically called a "covariant". We say that f is faithful if G acts faithfully on the image f(V). The covariant dimension of G is the minimum of the dimension of f(V) taken over all faithful covariants f. The essential dimension of G is defined in the same way, but allows for rational equivariant morphisms. The essential dimension and covariant dimension of G are related to cohomological invariants, generic polynomials and other topics, see the work of Buehler-Reichstein [BuR97]. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We also classify the groups of covariant dimension less or equal to 2. It turns out that they are the finite subgroups of GL(2,C). A byproduct of our investigations is the existence of a purely transcendental field of definition of degree n-3 for a generic field extension of degree n < 5.