Symmetric polynomials in the variety generated by Grassmann algebras

TL;DR

This paper investigates symmetric polynomials within the variety generated by Grassmann algebras, providing a generating set for the symmetric invariants in the commutator ideal of a free algebra in this variety.

Contribution

It introduces a free generating set for the symmetric invariants in the commutator ideal of the free algebra in the Grassmann algebra variety.

Findings

Identifies the structure of symmetric elements in the free algebra

Provides explicit generators for the symmetric invariants

Enhances understanding of polynomial identities in Grassmann algebra varieties

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 $(F_3')^{S_3}$.