| Abstract: |
| We show that a $q$-character of a Kirillov-Reshetikhin module (KR-modules) might be obtained from a specific cluster variable of a seed obtained by applying a maximal green sequence to the initial (infinite) quiver the Hernandez-Leclerc cluster algebra. For a collection of KR-modules with nested supports, we show an explicit construction of a cluster seed which has cluster variables corresponding to the $q$-characters of KR-modules of such a collection.
We prove that the product of KR-modules of such a collection is a simple module. We also construct cluster seeds with cluster variables corresponding to $q$-characters of KR-modules of some non-nested collections. We make a conjecture that tensor products of KR-modules for such non-nested collections are simple. We also show that the cluster Donaldson-Thomas transformations for double Bruhat cells can be computed the Frenkel-Mukhin algorithm and our algorithm. |
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