Fusion rules for quantum reflection groups

TL;DR

This paper determines the fusion rules for quantum analogues of complex reflection groups, providing explicit formulas for tensor products of irreducible representations, extending classical Clebsch-Gordan rules.

Contribution

It introduces the fusion rules for quantum complex reflection groups and describes the tensor product structure of their irreducible representations.

Findings

Irreducible representations indexed by elements of a free monoid

Explicit tensor product formulas similar to Clebsch-Gordan rules

Extension of classical representation theory to quantum groups

Abstract

We find the fusion rules for the quantum analogues of the complex reflection groups $H_n^s=\mathbb Z_s\wr S_n$. The irreducible representations can be indexed by the elements of the free monoid $\mathbb N^{*s}$, and their tensor products are given by formulae which remind the Clebsch-Gordan rules (which appear at $s=1$).