Schur-Weyl duality for tensor powers of the Burau representation

Artin’s braid group $B_n$ is generated by ${\sigma }_1,\dots ,{\sigma }_{n-1}$ subject to the relations $${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1},{\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i\text{ if }\mathrm{\mid }i-j\mathrm{\mid }>1.$$ For complex parameters $q_1,q_2$ such that $q_1{q}_2\mathrm{\ne }0$, the group $B_n$ acts on the vector space $$\mathbf{E}$ = \sum_i $\mathbb{C}$ $\mathbf{e}$_i$withbasis$$\mathbf{e}$_1, \dots, $\mathbf{e}$_n$ by \begin{gather*} \sigma_i \cdot $\mathbf{e}$i = (q_1+q_2)$\mathbf{e}$i + q_1$\mathbf{e}${i+1}, \quad \sigma_i \cdot $\mathbf{e}${i+1} = -q_2$\mathbf{e}$_i, \ \sigma_i \cdot $\mathbf{e}$_j = q_1 $\mathbf{e}$_j $\text{ if }$ j \ne i,i+1. \end{gather*} This representation is (a slight generalization of) the Burau representation. If $q = -q_2/q_1$ is not a root of unity, we show that the algebra of all endomorphisms of ${\mathbf{E}}^{\otimes r}$ commuting with the $B_n$-action is generated by the place-permutation action of the symmetric group $S_r$ and the operator $p_1$, given by $$p_1({\mathbf{e}}_{j_1}\otimes {\mathbf{e}}_{j_2}\otimes \cdots \otimes {\mathbf{e}}_{j_r})=q^{j_1-1}\sum _{i=1}^n{\mathbf{e}}_i\otimes {\mathbf{e}}_{j_2}\otimes \cdots \otimes {\mathbf{e}}_{j_r}\mathrm{.}$$ Equivalently, as a $(\mathbb{C}{B}_n,{\mathcal{P}}_r^{\mathrm{\prime }}([n{]}_q))$-bimodule, ${\mathbf{E}}^{\otimes r}$ satisfies Schur–Weyl duality, where ${\mathcal{P}}_r^{\mathrm{\prime }}([n{]}_q)$ is a certain subalgebra of the partition algebra ${\mathcal{P}}_r([n{]}_q)$ on $2r$ nodes with parameter $[n]_q = 1+q+\cdots + q^{n-1}$, isomorphic to the semigroup algebra of the “rook monoid” studied by W. D. Munn, L. Solomon, and others.