In 2021, Kashiwara--Kim--Oh--Park constructed cluster algebra structures on the Grothendieck rings of certain monoidal subcategories of the category of finite-dimensional representations of a quantum loop algebra, generalizing Hernandez--Leclerc’s pioneering work from 2010. They stated the problem of finding explicit exchange matrices for the seeds they used.
In the first part of the talk, we provide a solution by using Palu’s generalized mutation rule applied to the cluster categories associated with certain algebras of global dimension at most 2, for example tensor products of path algebras of representation-finite quivers. Our method is based on (and contributes to) the bridge, provided by cluster combinatorics, between the representation theory of quantum groups and that of quivers with relations.
In the second part of the talk, based on joint work with Qin and Wei, we show how the combinatorics of signed words can be translated into the language of i-boxes. Through this correspondence, we provide explicit formulas for the aforementioned exchange matrices, recovering those obtained by Kashiwara--Kim through the tools of monoidal categorification.