A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules

Abstract

We describe a cluster algebra algorithm for calculating $q$-characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. This yields a geometric $q$-character formula for tensor products of Kirillov–Reshetikhin modules. When $\mathfrak g$ is of type $A, D, E$, this formula extends Nakajima's formula for $q$-characters of standard modules in terms of homology of graded quiver varieties.