Three-term recurrence relations of minimal affinizations of type $G_2$

TL;DR

This paper introduces a simplified system of three-term recurrence relations, called the M-system, for minimal affinizations of type G2 in quantum affine algebras, linking it to cluster algebra structures.

Contribution

It presents the M-system of type G2, a new set of recurrence relations that simplifies previous extended T-systems and connects to cluster algebra exchange relations.

Findings

M-system contains all minimal affinizations of type G2

M-system is simpler than the extended T-system

Recurrence relations correspond to cluster algebra exchange relations

Abstract

Minimal affinizations form a class of modules of quantum affine algebras introduced by Chari. We introduce a system of equations satisfied by the $q$-characters of minimal affinizations of type $G_2$ which we call the M-system of type $G_2$. The M-system of type $G_2$ contains all minimal affinizations of type $G_2$ and only contains minimal affinizations. The equations in the M-system of type $G_2$ are three-term recurrence relations. The M-system of type $G_2$ is much simpler than the extended T-system of type $G_2$ obtained by Mukhin and the second author. We also interpret the three-term recurrence relations in the M-system of type $G_2$ as exchange relations in a cluster algebra constructed by Hernandez and Leclerc.