# A geometric $q$-character formula for snake modules

## TL;DR

This paper proves a geometric $q$-character formula for snake modules in certain quantum affine algebras, providing new combinatorial and algebraic insights into their structure and relation to cluster algebras.

## Contribution

It establishes the geometric $q$-character formula for snake modules in types A and B, and explores their cluster algebra properties and combinatorial formulas.

## Key findings

- Proved the geometric $q$-character formula for snake modules in types A and B.
- Showed snake modules correspond to cluster monomials with square-free denominators.
- Established factoriality of cluster algebras for types A, D, E.

## Abstract

Let $\mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of a simple complex Lie algebra ${\mathfrak{g}}$. Let $\mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. We prove the geometric $q$-character formula conjectured by Hernandez and Leclerc in types $\mathbb{A}$ and $\mathbb{B}$ for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the $F$-polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category $\mathscr{C}_1$ are factorial for Dynkin types $\mathbb{A,D,E}$.