Some special bases of the 2-swap algebras

TL;DR

This paper investigates the structure of a specific algebra generated by the symmetric group acting on a 2-dimensional tensor space, revealing that symmetric elements are spanned by involutions.

Contribution

It introduces a new understanding of the symmetric elements in 2-swap algebras, showing they are spanned by involutions of the symmetric group.

Findings

Symmetric elements are spanned by involutions.

The algebra is induced by the symmetric group action on a 2-dimensional tensor space.

Provides a basis description for symmetric elements in the algebra.

Abstract

We study the algebra $\Sigma_n$ induced by the action of the symmetric group $S_n$ on $V^{\otimes n}$ when $\dim V=2$. Our main result is that the space of symmetric elements of $\Sigma_n$ is linearly spanned by the involutions of $S_n$.