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 use a refinement of this considered first by Littlewood and later, by Sam and Snowde