Once More about Spectral-Dependent Quantum $R$-Matrix for $U_q(A_2)$

TL;DR

This paper explicitly computes spectral-dependent quantum R-matrices for certain representations of U_q(A_2), revealing a novel dependence on q in fractional powers, which is a new feature in the literature.

Contribution

It provides the first explicit example of an R-matrix with fractional powers of q dependence for U_q(A_2) representations.

Findings

Explicit spectral-dependent R-matrices computed for specific representations.

Discovery of R-matrix dependence on q in fractional powers.

First known example of such a feature in literature.

Abstract

The recently obtained results in \cite{ZG2} are used to compute the explicitly spectral-dependent $R$-matrix (or the intertwiners) on $V_{(6)}(x)\otimes V_{(6)}(y)$ and $V_{(3)}(x)\otimes V_{(6)}(y)$, where $V_{(6)}$ and $V_{(3)}$ are the 6-dimensional and fundamental representations of $U_q(A_2)$, respectively. It appears that the $R$-matrix on $V_{(3)}(x)\otimes V_{(6)}(y)$ depends on $q$ in the different way from what one might usually think: $q$ occurs in the $R$-matrix in fractional powers. It seems to be the first example in literatures of $R$-matrix with the new feature.