Projection formulas and a refinement of Schur--Weyl--Jones duality for symmetric groups

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.