We show that the nβth symmetric product of an affine scheme
X=SpecA over a characteristic zero field is isomorphic as a scheme
to the quotient by the general linear group of the scheme parameterizing
nβdimensional linear representations of A. As a consequence we give
generators and relations of the related rings of invariants as well as the
equations of any symmetric products in term of traces. In positive
characteristic we prove an analogous result for the associated varieties.Comment: 10 pages, from the talk given at ICM 2006 Madri