Integral Schur-Weyl duality for partition algebras

TL;DR

This paper establishes an integral Schur-Weyl duality for tensor spaces over a commutative ring, linking the actions of the Weyl group algebra and partition algebra, extending classical duality results.

Contribution

It proves the Schur-Weyl duality for tensor space over a ring with partition algebras, including the half partition algebra, broadening the duality's applicability.

Findings

Tensor space satisfies Schur-Weyl duality over a ring.

Duality involves the Weyl group algebra and partition algebra actions.

Results extend classical duality to integral and algebraic settings.

Abstract

Let $V$ be a free module of rank $n$ over a commutative unital ring $k$. We prove that tensor space $V^{\otimes r}$ satisfies Schur--Weyl duality, regarded as a bimodule for the action of the group algebra of the Weyl group of $\mathrm{GL}(V)$ and the partition algebra $P_r(n)$ over $k$. We also prove a similar result for the half partition algebra.