Cluster algebras and snake modules

TL;DR

This paper proves the Hernandez-Leclerc conjecture for prime snake modules of types A_n and B_n by establishing their correspondence with cluster variables and showing they satisfy specific equations linked to cluster algebra mutations.

Contribution

It demonstrates that prime snake modules are real and establishes their correspondence with cluster variables, confirming the Hernandez-Leclerc conjecture for these types.

Findings

Prime snake modules are real.

Equations in the S-system correspond to cluster mutations.

Prime snake modules correspond to cluster variables.

Abstract

Snake modules introduced by Mukhin and Young form a family of modules of quantum affine algebras. The aim of this paper is to prove that the Hernandez-Leclerc conjecture about monoidal categorifications of cluster algebras is true for prime snake modules of types $A_{n}$ and $B_{n}$. We prove that prime snake modules are real. We introduce $S$-systems consisting of equations satisfied by the $q$-characters of prime snake modules of types $A_{n}$ and $B_{n}$. Moreover, we show that every equation in the $S$-system of type $A_n$ (respectively, $B_n$) corresponds to a mutation in the cluster algebra $\mathscr{A}$ (respectively, $\mathscr{A}'$) constructed by Hernandez and Leclerc and every prime snake module of type $A_n$ (respectively, $B_n$) corresponds to some cluster variable in $\mathscr{A}$ (respectively, $\mathscr{A}'$). In particular, this proves that the Hernandez-Leclerc conjecture is true for all prime snake modules of types $A_{n}$ and $B_{n}$.