On the minimal affinizations over the quantum affine algebras of type $C_n$

TL;DR

This paper explores the relationship between minimal affinizations of quantum affine algebras of type C_n and cluster algebras, revealing that their q-characters satisfy mutation equations and correspond to cluster variables.

Contribution

It establishes a connection between minimal affinizations and cluster algebra structures for type C_n quantum affine algebras, using q-character systems.

Findings

q-characters satisfy specific mutation equations

Minimal affinizations correspond to cluster variables

Cluster algebra framework describes minimal affinizations

Abstract

In this paper, we study the minimal affinizations over the quantum affine algebras of type $C_n$ by using the theory of cluster algebras. We show that the $q$-characters of a large family of minimal affinizations of type $C_n$ satisfy some systems of equations. These equations correspond to mutation equations of some cluster algebras. Furthermore, we show that the minimal affinizations in these equations correspond to cluster variables in these cluster algebras.