Generalized symmetric functions and invariants of matrices

We generalize the classical isomorphism between symmetric functions and invariants of a matrix. In particular, we show that the invariants over several matrices are given by the abelianization of the symmetric tensors over the free associative algebra. The main result is proved by finding a characteristic free presentation of the algebra of symmetric tensors over a free algebra. The author is supported by research grant Politecnico di Torino n.119, 2004.