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We show the denominator formulas for the normalized $R$-matrix involving twoarbitrary 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 alsoconjecture a uniform denominator formulas for all simply-laced types; inparticular, type $E_n^{(1)}$. With the denominator formulas, we determine thesimplicity 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 setsof KR modules for the untwisted affine types, suggested by Hernandez andLeclerc as clusters, form strongly commuting families, which implies that allcluster monomials in the clusters are real simple modules.

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