Symmetric polynomials in the variety generated by Grassmann algebras

Let G denote the variety generated by infinite-dimensional Grassmann algebras, i.e. the collection of all unitary associative algebras satisfying the identity [[z1, z2], z3] = 0, where [zi, zj] = zizj - zjzi. Consider the free algebra F3 in G generated by X3 = {x1, x2, x3}. We call a polynomial p. F3 symmetric if it is preserved under the action of the symmetric group S3 on generators, i.e. p(x1, x2, x3) = p(x.1, x.2, x.3) for each permutation.. S3. The set of symmetric polynomials forms the subalgebra F S3 3 of invariants of the group S3 in F3. The commutator ideal F x 3 of the algebra F3 has a natural left K[X3]-module structure, and (F x 3)S3 is a left K[X3]S3 -module. We give a finite free generating set for the K[X3]S3 -module (F x 3)S3