# Schur-Weyl duality for $U_{v,t}(sl_{n})$

## TL;DR

This paper develops a Schur-Weyl duality framework for the two-parameter quantum algebra $U_{v,t}(sl_{n})$, constructing an R-matrix and explicit irreducible representations, advancing understanding of quantum symmetries.

## Contribution

It introduces a Hopf pairing to realize $U_{v,t}(sl_{n})$ as a Drinfeld double and constructs explicit irreducible representations and primitive idempotents.

## Key findings

- Constructed an R-matrix for $U_{v,t}(sl_{n})$
- Established Schur-Weyl duality with Hecke algebra $H_{k}(v,t)$
- Explicitly constructed irreducible $U_{v,t}(sl_{n})$-representations

## Abstract

In \cite{fl}, the authors get a new presentation of two-parameter quantum algebra $U_{v,t}(\mathfrak{g})$. Their presentation can cover all Kac-Moody cases. In this paper, we construct a suitable Hopf pairing such that $U_{v,t}(sl_{n})$ can be realized as Drinfeld double of certain Hopf subalgebras with respect to the Hopf pairing. Using Hopf pairing, we construct a $R$-matrix for $U_{v,t}(sl_{n})$ which will be used to give the Schur-Weyl dual between $U_{v,t}(sl_{n})$ and Hecke algebra $H_{k}(v,t)$. Furthermore, using the Fusion procedure we construct the primitive orthogonal idempotents of $H_{k}(v,t)$. As a corollary, we give the explicit construction of irreducible $U_{v,t}(sl_{n})$-representations of $V^{\otimes k}$.