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 ${\left({\mathbb{C}}^n\right)}^{\otimes k}\mathrm{.}$ 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 $S_k\mathrm{.}$ We use this refinement to construct subspaces of ${\left({\mathbb{C}}^n\right)}^{\otimes k}$ that are isomorphic to certain irreducible representations of $S_n\times S_k\mathrm{.}$ 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.