Selection and identity rules for subductions of type A quantum Iwahori-Hecke algebras

TL;DR

This paper develops combinatorial rules and criteria for decomposing irreducible representations of type A quantum Iwahori-Hecke algebras into subalgebra components, extending classical symmetric group results to the quantum setting.

Contribution

It introduces a combinatorial description, a selection rule based on the Richardson-Littlewood criterion, and an equivariance condition for subduction coefficients in quantum Iwahori-Hecke algebras.

Findings

Provides a criterion to determine vanishing coupling coefficients.

Extends classical symmetric group subduction results to quantum algebras.

Offers a combinatorial framework for the subduction problem.

Abstract

This paper is concerned with the subduction problem of type A quantum Iwahori-Hecke algebras $\mathbb{C} \mathbf{H}(\mathfrak{S}_f,q^2)$ with a real deformation parameter $q$, i.e. the problem of decomposing irreducible representations of such algebras as direct sum of irreducible representations of the subalgebras $\mathbb{C}\mathbf{H}(\mathfrak{S}_{f_1}, q^2) \times \mathbb{C}\mathbf{H}(\mathfrak{S}_{f_2}, q^2)$, with $f_1 + f_2 = f$. After giving a suitable combinatorial description for the subduction issue, we provide a selection rule, based on the Richardson-Littlewood criterion, which allows to determine the vanishing coupling coefficients between standard basis vectors for such representations, and we also present an equivariance condition for the subduction coefficients. Such results extend those ones corresponding to the subduction problem in symmetric group algebras $\mathbb{C}\mathfrak{S}_f \downarrow \mathbb{C}\mathfrak{S}_{f_1} \times \mathbb{C} \mathfrak{S}_{f_2}$ which are obtained by $q$ approaching the value 1.