Schur-Weyl duality for tensor powers of the Burau representation (2012.10439v3)

Abstract: 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}, \quad \sigma_i\sigma_j = \sigma_j \sigma_i \text{ if } |i-j|>1. ] For complex parameters $q_1,q_2$ such that $q_1q_2 \ne 0$, the group $B_n$ acts on the vector space $\mathbf{E} = \sum_i \mathbb{C} \mathbf{e}i$ with basis $\mathbf{e}_1, \dots, \mathbf{e}_n$ 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} . ] Equivalently, as a $(\mathbb{C} B_n, \mathcal{P}'_r([n]_q))$-bimodule, $\mathbf{E}^{\otimes r}$ satisfies Schur--Weyl duality, where $\mathcal{P}'_r([n]_q)$ 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.