Symmetric polynomials in free associative algebras

By a result of Margarete Wolf in 1936, we know that the algebra K < X-d >(Sym(d)) of symmetric polynomials in noncommuting variables is not finitely generated. In 1984, Koryukin proved that if we equip the homogeneous component of degree n with the additional action of Sym(n) by permuting the positions of the variables, then the algebra of invariants K < X-d >(G) of every reductive group G is finite generated. First, we make a short comparison between classical invariant theory of finite groups and its noncommutative counterpart. Then, we expose briefly the results of Wolf. Finally, we present the main result of our paper, which is, over a field of characteristic 0 or of characteristic p > d, the algebra K < X-d >(Sym(d)) with the action of Koryukin is generated by the elementary symmetric polynomials.