Products of Kirillov–Reshetikhin modules and maximal green sequences

Abstract We show that the $q$-character of a Kirillov–Reshetikhin module (KR module) for untwisted quantum affine algebras of simply laced types $A_{n}^{(1)}$, $D_{n}^{(1)}$, $E_{6}^{(1)}$, $E_{7}^{(1)}$, and $E_{8}^{(1)}$ can be obtained from a specific cluster variable in a seed obtained by applying a maximal green sequence to the initial (infinite) quiver of the Hernandez–Leclerc cluster algebra. For a collection of KR modules with nested supports, we provide an explicit construction of a cluster seed whose cluster variables correspond to the $q$-characters of the KR modules in the collection. We prove that the tensor product of KR modules in such a collection forms a simple module. We also construct cluster seeds with cluster variables corresponding to the $q$-characters of KR modules from certain non-nested collections. Finally, we show that the cluster Donaldson–Thomas transformations for double Bruhat cells of $ADE$ types can be computed using the $q$-characters of KR modules.