Solution of a Problem in Monoidal Categorification by Additive Categorification

TL;DR

This paper advances monoidal categorification by explicitly constructing quivers for cluster algebra structures on quantum group representations, bridging quantum groups and quiver representations through cluster combinatorics.

Contribution

It provides an explicit method to find quivers for cluster seeds in monoidal categorification using Palu's mutation rule and cluster categories.

Findings

Explicit quivers constructed for cluster algebra structures

Application of Palu's mutation rule to categories of global dimension ≤ 2

Strengthening the link between quantum groups and quiver representations

Abstract

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 quivers for the seeds they used. 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. Thus, 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.