Construction of $R$-matrices for symmetric tensor representations related to $U_{q}(\widehat{sl_{n}})$

TL;DR

This paper presents a new explicit factorized form of the $R$-matrix for symmetric tensor representations of the quantum affine algebra $U_q(\\widehat{sl_n})$, including proofs of stochasticity and symmetry properties.

Contribution

It introduces a novel factorized representation of the $R$-matrix for symmetric tensor representations using a 3D approach, with explicit formulas and symmetry analysis.

Findings

Explicit formulas for $R$-matrix elements derived

Proof that the twisted $R$-matrix is stochastic

Discussion of symmetries and degenerations of the $R$-matrix

Abstract

In this paper we construct a new factorized representation of the $R$-matrix related to the affine algebra $U_{q}(\widehat{sl_{n}})$ for symmetric tensor representations with arbitrary weights. Using the 3D approach we obtain explicit formulas for the matrix elements of the $R$-matrix and give a simple proof that a "twisted" $R$-matrix is stochastic. We also discuss symmetries of the $R$-matrix, its degenerations and compare our formulas with other results available in the literature.