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

Schur-Weyl duality concerns the actions of GL₍ (C) and S₊ on tensor powers of the form V^ 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 GL₍ (C) to subgroups of GL₍ (C). This leads us to consider further variants of Schur-Weyl duality, with the use of variants of the tensor space V^ k. Instead of taking repeated tensor products of V, we make use of a freest commutative bilinear operation in place of, and this is motivated by an associated invariance property given by the action of S₊. By then taking the centralizer algebra with respect to the action of the group of permutation matrices in GL₍ (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.