On the minimal affinizations of type $F_4$

TL;DR

This paper explores the application of cluster algebra theory to minimal affinizations in quantum affine algebra of type F4, establishing equations for their q-characters and introducing dominant monomial graphs.

Contribution

It connects minimal affinizations of type F4 with cluster variables and proposes conjectural equations for those outside the main system, advancing understanding of their algebraic structure.

Findings

q-characters satisfy a system of equations

Minimal affinizations correspond to cluster variables

Introduction of dominant monomial graphs for analysis

Abstract

In this paper, we apply the theory of cluster algebras to study minimal affinizations for the quantum affine algebra of type $F_4$. We show that the $q$-characters of a large family of minimal affinizations of type $F_4$ satisfy a system of equations. Moreover, a minimal affinization in this system corresponds to some cluster variable in some cluster algebra $\mathscr{A}$. For the other minimal affinizations of type $F_4$ which are not in this system, we give some conjectural equations which contains these minimal affinizations. Furthermore, we introduce the concept of dominant monomial graphs to study the equations satisfied by $q$-characters of modules of quantum affine algebras.