Generalized symmetric functions

It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on matrices that are invariants by the action of conjugation by general linear group. We generalize this result showing that the abelianization of the algebra of the symmetric tensors of fixed order over a free associative algebra is isomorphic to the algebra of the polynomials invariants of several matrices over an infinite field or the integers. While proving the main result we find generators and relations of abelianized divided powers of an algebra over any commutative ring.