Symmetric polynomials in the variety generated by Grassmann algebras

Abstract

Let $\mathcal{G}$ denote the variety generated by infinite dimensional Grassmann algebras; i.e., the collection of all unitary associative algebras satisfying the identity $[[z_1,z_2],z_3]=0$, where $[z_i,z_j]=z_iz_j-z_jz_i$. Consider the free algebra $F_3$ in $\mathcal{G}$ generated by $X_3=\{x_1,x_2,x_3\}$. The commutator ideal $F_3'$ of the algebra $F_3$ has a natural $K[X_3]$-module structure. We call an element $p\in F_3$ symmetric if $p(x_1,x_2,x_3)=p(x_{\xi1},x_{\xi2},x_{\xi3})$ for each permutation $\xi\in S_3$. Symmetric elements form the subalgebra $F_3^{S_3}$ of invariants of the symmetric group $S_3$ in $F_3$. We give a free generating set for the $K[X_3]^{S_3}$-module $(F_3')^{S_3}$.