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

Abstract

We show the denominator formulas for the normalized $R$-matrix involving two arbitrary Kirillov--Reshetikhin (KR) modules $W_{m,a}^{(k)}$ and $W_{p,b}^{(l)}$ 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.

This paper has been superseded by arXiv:2510.10874. The previous paper arXiv:1910.10347, which has not been published elsewhere, was written by the same authors. The new version extends the previous work by proving many of the earlier conjectures and presenting several new results. Therefore, the earlier version has been withdrawn and replaced with this updated one