Schur–Weyl Reciprocity between the Quantum Superalgebra and the Iwahori–Hecke Algebra

Abstract

In this paper, we establish Schur–Weyl reciprocity between the quantum general super Lie algebra $U_q^\sigma\big{(}{\mathfrak{g}\mathfrak{l}}(m,n)\big{)}$ and the Iwahori–Hecke algebra $\mathcal{H}_{\mathbb{Q}(q),r}(q)$. We introduce the sign $q$-permutation representation of $\mathcal{H}_{\mathbb{Q}(q),r}(q)$ on the tensor space $V^{{\otimes}r}$ of $(m+n)$ dimensional $\mathbb{Z}_{\,2}$-graded $\mathbb{Q}(q)$-vector space $V=V_{\bar{0}}{\oplus}V_{\bar{1}}$. This action commutes with that of $U_q^\sigma\big{(}{\mathfrak{g}\mathfrak{l}}(m,n)\big{)}$ derived from the vector representation on $V$. Those two subalgebras of $\operatorname{End}_{\mathbb{Q}(q)}(V^{{\otimes}r})$ satisfy Schur–Weyl reciprocity. As special cases, we obtain the super case ($q{\rightarrow}1$), and the quantum case ($n=0$). Hence this result includes both the super case and the quantum case, and unifies those two important cases.