  Schur--Weyl--Jones duality establishes the connection between the commuting
actions of the symmetric group $S_{n}$ and the partition algebra $P_{k}(n)$ on
the tensor space $\left(\mathbb{C}^n\right)^{\otimes k}.$ We give a refinement
of this, determining a subspace of $\left(\mathbb{C}^n\right)^{\otimes k}$ on
which we have a version of Schur--Weyl duality for the symmetric groups $S_{n}$
and $S_{k}.$
  We use this refinement to construct subspaces of
$\left(\mathbb{C}^n\right)^{\otimes k}$ that are isomorphic to certain
irreducible representations of $S_{n}\times S_{k}.$ We then use the Weingarten
calculus for the symmetric group to obtain an explicit formula for the
orthogonal projection from $\left(\mathbb{C}^n\right)^{\otimes k}$ to each
subspace.
