Schur–Weyl duality for tensor powers of the Burau representation

Artin’s braid group $$B_n$$ is generated by $$\sigma _1, \ldots , \sigma _{n-1}$$ subject to the relations $$\begin{aligned} _i _{i+1} = _{i+1}, \quad _i\sigma _j \text { if } |i-j|>1. \end{aligned}$$ For complex parameters $$q_1,q_2$$ such that $$q_1q_2 \ne 0$$ acts on vector space $$\mathbf {E}= \sum \mathbb {C}\mathbf {e}_i$$ with basis {e}_1, \mathbf {e}_n$$ \cdot {e}_i= & {} (q_1+q_2)\mathbf {e}_i + q_1\mathbf {e}_{i+1}, {e}_{i+1} -q_2\mathbf {e}_i, \\ {e}_j= q_1 {e}_j }\; j i,i+1. This representation (a slight generalization of) Burau representation. If $$q -q_2/q_1$$ not a root of unity, we show algebra all endomorphisms {E}^{\otimes r}$$ commuting -action place-permutation action symmetric $$S_r$$ and operator $$p_1$$ given p_1(\mathbf {e}_{j_1} \otimes {e}_{j_2} \cdots {e}_{j_r}) q^{j_1-1} \, _{i=1}^n {e}_{j_r} . Equivalently, as $$(\mathbb {C}B_n, \mathcal {P}'_r([n]_q))$$ -bimodule, satisfies Schur–Weyl duality, where $$\mathcal {P}'_r([n]_q)$$ certain subalgebra partition {P}_r([n]_q)$$ 2r nodes parameter $$[n]_q 1+q+\cdots q^{n-1}$$ isomorphic semigroup “rook monoid” studied W. D. Munn, L. Solomon, others.