Symmetric products, linear representations and the commuting scheme

Abstract

We show that the ring of multisymmetric functions over a commutative ring is isomorphic to the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices. As a consequence we give a surjection from the ring of invariants of several matrices to the ring of multisymmetric functions generalizing a classical result of H.Weyl and F.Junker. We also find a surjection from the ring of invariants over the commuting scheme to the ring of multisymmetric functions. This surjection is an isomophism over a characteristic zero field and induces an isomorphism at the level of reduced structures over an infinite field of positive characteristic.