Séminaires : Séminaire d'Algèbre

We show that a qq-character of a Kirillov-Reshetikhin module (KR modules) for untwisted quantum affine algebras of simply laced types $A_n^{(1)}$​,  $D_n^{(1)}$​,  $E_6^{(1)}$​,  $E_7^{(1)}$​,  $E_8^{(1)}$​, might be obtained from a specific cluster variable of 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 show an explicit construction of a cluster seed, which has cluster variables corresponding to the qq-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 have an explicit construction of cluster seeds with cluster variables corresponding to qq-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. If time permits, I explain that the cluster Donaldson-Thomas transformations for double Bruhat cells for ADE types can be computed using qq-characters of KR-modules, and new algorithm to compute qq-characters KR-modules. The talk is based on joint work with Y.Kanakubo and T.Nakashima.

This talk will take place in hybrid mode at the Institut Henri Poincaré.