Schur-Weyl Reciprocity for the Hecke Algebra of $(\Bbb Z/r\Bbb Z)\wr \frak S_n$

TL;DR

This paper establishes a reciprocity between a subalgebra of the quantum group $U_q(gl_r)$ and the Hecke algebra of the wreath product $(Z/rZ) times S_n$, including their mutual commutant and irreducible decompositions.

Contribution

It introduces a new reciprocity relation between $U_q(h)$ and the Hecke algebra $ ext{Hecke}_{n,r}$, extending Schur-Weyl duality to this setting.

Findings

The commutant of $U_q(h)$ is isomorphic to a quotient of $ ext{Hecke}_{n,r}$.

The irreducible decomposition of tensor powers under $ ext{Hecke}_{n,r}$ is explicitly determined.

A reciprocity between $U_q(h)$ and $ ext{Hecke}_{n,r}$ is established.

Abstract

The purpose of this paper is to give a reciprocity between $U_q(h)$ and $\Cal H_{n,r}$, the Hecke algebra of $(\Bbb Z / r\Bbb Z)\wr \frak S_n$ introduced by Ariki and Koike. Let $K=\Bbb Q(q,u_1,\dots ,u_r)$ 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 $U_q(gl_r)$ and the Hecke algebra $\Cal H_n$. We denote by $U_q(h)$ the $K$-subalgebra of $U_q(gl_r)$ 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 $\Cal H_{n,r}$.