Schur-Weyl Reciprocity for the Hecke Algebra of $(\mathbb{Z}/r\mathbb{Z})\wr {\mathfrak{S}}_n$

Abstract

The purpose of this paper is to give a reciprocity between $U_q(h)$ and $\Cal H_{n,r}$, the Hecke algebra of $(\mathbb{Z}/r\mathbb{Z})\wr {\mathfrak{S}}_n$ introduced by Ariki and Koike. Let $K=\mathbb{Q}(q,u_1,\dots ,u_r)$ be the field of rational funcitons in variables $q,u_1,\dots ,u_r$. We adopt $K$ as the base field for both the quantized universal enveloping algebra $U_q(g{l}_r)$ and the Hecke algebra $\Cal H_n$. We denote by $U_q(h)$ the $K$-subalgebra of $U_q(g{l}_r)$ generated by $q^{E_{ii}}$'s $(1\le i\le r)$. In this paper, we show that the commutant of $U_q(h)$ in $End((K^r{)}^{\otimes n})$ is isomorphic to a quotient of $\Cal H_{n,r}$. We also determine the irreducible decomposition of $(K^r{)}^{\otimes n}$ under the action of $\Cal H_{n,r}$. As a consequence, we obtain the reciprocity for $U_q(h)$ and $\Cal H_{n,r}$.

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