Hernandez-Leclerc modules and snake graphs

TL;DR

This paper introduces a new combinatorial approach using snake graphs to compute the $q$-characters of Hernandez-Leclerc modules, connecting quantum affine algebra representations with cluster algebra structures.

Contribution

It provides a non-recursive formula for $q$-characters of Hernandez-Leclerc modules via snake graphs and an explicit formula for $F$-polynomials, advancing the understanding of their structure.

Findings

Explicit non-recursive $q$-character formula using snake graphs

New recursive formula for $q$-characters

Connection between Hernandez-Leclerc modules and cluster algebra combinatorics

Abstract

In 2010, Hernandez and Leclerc studied connections between representations of quantum affine algebras and cluster algebras. In 2019, Brito and Chari defined a family of modules over quantum affine algebras, called Hernandez-Leclerc modules. We characterize the highest $\ell$-weight monomials of Hernandez-Leclerc modules. We give a non-recursive formula for $q$-characters of Hernandez-Leclerc modules using snake graphs, which involves an explicit formula for $F$-polynomials. We also give a new recursive formula for $q$-characters of Hernandez-Leclerc modules.