Simplicity of tensor products of Kirillov--Reshetikhin modules: nonexceptional affine and G types

TL;DR

This paper derives denominator formulas for normalized R-matrices involving Kirillov--Reshetikhin modules across various affine types, establishing their simplicity, pole degrees, and cluster properties, thus advancing understanding of tensor products in quantum affine algebras.

Contribution

It provides explicit denominator formulas for KR modules in all nonexceptional affine types, including new results for D4^{(3)} and G_2^{(1)}, and proves their implications for cluster monomials.

Findings

Denominator formulas for all nonexceptional affine types, D4^{(3)}, and G_2^{(1).

Complete characterization of tensor product simplicity and R-matrix pole degrees.

Proof that certain KR modules form strongly commuting families, confirming cluster monomials are real simple modules.

Abstract

We show the denominator formulas for the normalized $R$-matrix involving two arbitrary Kirillov--Reshetikhin (KR) modules $W^{(k)}_{m,a}$ and $W^{(l)}_{p,b}$ in all nonexceptional affine types, $D_4^{(3)}$, and $G_2^{(1)}$. To achieve our goal, we prove the existence of homomorphisms, which can be understood as generalization of Dorey rule to KR modules. We also conjecture a uniform denominator formulas for all simply-laced types; in particular, type $E_n^{(1)}$. With the denominator formulas, we determine the simplicity of tensor product of KR modules and degrees of poles of normalized $R$-matrices between two KR modules completely in nonexceptional affine types, $D_4^{(3)}$, and $G_2^{(1)}$. As an application, we prove that the certain sets of KR modules for the untwisted affine types, suggested by Hernandez and Leclerc as clusters, form strongly commuting families, which implies that all cluster monomials in the clusters are real simple modules.