A Schur-Weyl duality analogue based on a commutative bilinear operation

Schur-Weyl duality concerns the actions of $\text{GL}_{n}(\mathbb{C})$ and $S_{k}$ on tensor powers of the form $V^{\otimes k}$ for an $n$-dimensional vector space $V$. There are rich histories within representation theory, combinatorics, and statistical mechanics involving the study and use of diagram algebras, which arise through the restriction of the action of $\text{GL}_{n}(\mathbb{C})$ to subgroups of $\text{GL}_{n}(\mathbb{C})$. This leads us to consider further variants of Schur-Weyl duality, with the use of variants of the tensor space $V^{\otimes k}$. Instead of taking repeated tensor products of $V$, we make use of a freest commutative bilinear operation in place of $\otimes$, and this is motivated by an associated invariance property given by the action of $S_{k}$. By then taking the centralizer algebra with respect to the action of the group of permutation matrices in $\text{GL}_{n}(\mathbb{C})$, this gives rise to a diagram-like algebra spanned by a new class of combinatorial objects. We construct orbit-type bases for the centralizer algebras introduced in this paper, and we apply these bases to prove a combinatorial formula for the dimensions of our centralizer algebras.