  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.

Representation Theory