Schur–Weyl duality for tensor powers of the Burau representation

Abstract

Artin’s braid group \(B_n\) is generated by \(\sigma _1, \ldots , \sigma _{n-1}\) subject to the relations $$\begin{aligned} \sigma _i \sigma _{i+1} \sigma _i = \sigma _{i+1} \sigma _i \sigma _{i+1}, \quad \sigma _i\sigma _j = \sigma _j \sigma _i \text { if } |i-j|>1. \end{aligned}$$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 \(\mathbf {e}_1, \ldots , \mathbf {e}_n\) by $$\begin{aligned} \sigma _i \cdot \mathbf {e}_i= & {} (q_1+q_2)\mathbf {e}_i + q_1\mathbf {e}_{i+1}, \quad \sigma _i \cdot \mathbf {e}_{i+1} = -q_2\mathbf {e}_i, \\ \sigma _i \cdot \mathbf {e}_j= & {} q_1 \mathbf {e}_j \quad \text { if }\; j \ne i,i+1. \end{aligned}$$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 $$\begin{aligned} p_1(\mathbf {e}_{j_1} \otimes \mathbf {e}_{j_2} \otimes \cdots \otimes \mathbf {e}_{j_r}) = q^{j_1-1} \, \sum _{i=1}^n \mathbf {e}_i \otimes \mathbf {e}_{j_2} \otimes \cdots \otimes \mathbf {e}_{j_r} . \end{aligned}$$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.