Products of Kirillov-Reshetikhin modules and maximal green sequences

TL;DR

This work develops a precise bridge between $q$-characters of Kirillov-Reshetikhin modules for simply laced untwisted quantum affine algebras and Hernandez-Leclerc cluster algebras via maximal green sequences. It constructs explicit seeds whose cluster variables encode $q$-characters of KR-modules, proves simplicity of tensor products for nested collections, and analyzes non-nested cases with conjectures on simplicity. It further links cluster Donaldson-Thomas transformations to $q$-character data and provides efficient computational approaches using Frenkel-Mukhin or Nakajima algorithms, including extensions to infinite quivers and double Bruhat cells. The results yield a coherent framework to compute and interpret $q$-characters within cluster algebra mutation dynamics, with potential impact on representation theory and cluster-geometry applications.

Abstract

We show that a $q$-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 $q$-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 show that the cluster Donaldson-Thomas transformations for double Bruhat cells for $ADE$ types can be computed using $q$-characters of KR-modules.

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Theorems & Definitions (45)

• Remark 2.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Remark 3.4
• Definition 3.5
• Definition 3.6
• Proposition 3.7
• Example 3.8
• Proposition 3.9