Let [Formula: see text] denote the variety generated by infinite-dimensional Grassmann algebras, i.e. the collection of all unitary associative algebras satisfying the identity [Formula: see text], where [Formula: see text]. Consider the free algebra [Formula: see text] in [Formula: see text] generated by [Formula: see text]. We call a polynomial [Formula: see text] symmetric if it is preserved under the action of the symmetric group [Formula: see text] on generators, i.e. [Formula: see text] for each permutation [Formula: see text]. The set of symmetric polynomials forms the subalgebra [Formula: see text] of invariants of the group [Formula: see text] in [Formula: see text]. The commutator ideal [Formula: see text] of the algebra [Formula: see text] has a natural left [Formula: see text]-module structure, and [Formula: see text] is a left [Formula: see text]-module. We give a finite free generating set for the [Formula: see text]-module [Formula: see text].
On the symmetric polynomials in the variety of Grassmann algebras
