Research in the Mathematical Sciences ( IF 1.2 ) Pub Date : 2021-07-20 , DOI: 10.1007/s40687-021-00282-3 Stephen Doty 1 , Anthony Giaquinto 1
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.
中文翻译：
Burau 表示的张量幂的 Schur-Weyl 对偶性
Artin 的辫群\(B_n\)由\(\sigma _1, \ldots , \sigma _{n-1}\)产生
$$\begin{对齐} \sigma _i \sigma _{i+1} \sigma _i = \sigma _{i+1} \sigma _i \sigma _{i+1}, \quad \sigma _i\sigma _j = \sigma _j \sigma _i \text { if } |ij|>1。\end{对齐}$$对于复参数\(q_1,q_2\)使得\(q_1q_2 \ne 0\)，群\(B_n\)作用于向量空间\(\mathbf {E}= \sum _i \mathbb {C}\ mathbf {e}_i\)以\(\mathbf {e}_1, \ldots , \mathbf {e}_n\)为基础
$$\begin{aligned} \sigma _i \cdot \mathbf {e}_i= & {} (q_1+q_2)\mathbf {e}_i + q_1\mathbf {e}_{i+1}, \quad \ sigma _i \cdot \mathbf {e}_{i+1} = -q_2\mathbf {e}_i, \\ \sigma _i \cdot \mathbf {e}_j= & {} q_1 \mathbf {e}_j \四元 \text { 如果 }\; j \ne i,i+1。\end{对齐}$$这种表示是（略微概括）Burau 表示。如果\(q = -q_2/q_1\)不是统一根，我们证明\(\mathbf {E}^{\otimes r}\)的所有自同态的代数与\(B_n\) -action 由对称群\(S_r\)和算子\(p_1\)的位置置换动作生成，由下式给出
$$\begin{aligned} p_1(\mathbf {e}_{j_1} \otimes \mathbf {e}_{j_2} \otimes \cdots \otimes \mathbf {e}_{j_r}) = q^{j_1 -1} \, \sum _{i=1}^n \mathbf {e}_i \otimes \mathbf {e}_{j_2} \otimes \cdots \otimes \mathbf {e}_{j_r} 。\end{对齐}$$等价地，作为\((\mathbb {C}B_n, \mathcal {P}'_r([n]_q))\) -bimodule, \(\mathbf {E}^{\otimes r}\)满足 Schur –Weyl对偶性，其中\(\mathcal {P}'_r([n]_q)\)是2 r个节点上的分区代数\(\mathcal {P}_r([n]_q)\)的某个子代数参数为\([n]_q = 1+q+\cdots + q^{n-1}\)，同构于 W. D. Munn、L. Solomon 等人研究的“车幺半群”的半群代数。
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