Cluster algebras and snake modules

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 $\mathcal{A}$ (respectively, ${\mathcal{A}}^{\prime }$) constructed by Hernandez and Leclerc and every prime snake module of type $A_n$ (respectively, $B_n$) corresponds to some cluster variable in $\mathcal{A}$ (respectively, ${\mathcal{A}}^{\prime }$). In particular, this proves that the Hernandez-Leclerc conjecture is true for all prime snake modules of types $A_n$ and $B_n$.

References in corpus (3)

• Triangular bases in quantum cluster algebras and monoidal categorification conjectures
• M-systems and Cluster algebras
• Every finite acyclic quiver is a full subquiver of a quiver mutation equivalent to a bipartite quiver

Cited by in corpus (1)

• On the properties of the density matrix of the ${\mathfrak{sl}}_{n+1}$-invariant model