On the simplicity of the tensor product of two simple modules of quantum affine algebras

TL;DR

This paper extends criteria for the irreducibility of tensor products of simple modules in quantum affine algebras, providing new conditions applicable for all ranks and connecting to cluster algebra compatibility.

Contribution

It generalizes existing irreducibility criteria to all ranks and establishes a link between module tensor products and cluster algebra compatibility.

Findings

Criteria proven for snake modules and fundamental modules at extremity nodes

Compatibility of ladders characterized by tableaux satisfying the criterion

Generalization of weak separation condition in cluster algebras

Abstract

Lapid and M\'{i}nguez gave a criterion of the irreducibility of the parabolic induction $\sigma \times \pi$, where $\sigma$ is a ladder representation and $\pi$ is an arbitrary irreducible representation of the general linear group over a non-archimedean field. Through quantum affine Schur-Weyl duality, when $k$ is large enough, this gives a criterion of the irreducibility of the tensor product of a snake module $L(M)$ and any simple module $L(N)$ of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}_k})$. The goal of this paper is to add conditions to their criterion such that it works for any $k \geq 1$. We prove the criterion in the case where both modules are snake modules or one of them is a fundamental module at an extremity node and the other is any simple module. We also defined a similar criterion in the Grassmannian cluster algebra $\mathbb{C}[\mathrm{Gr}(k,n, \sim)]$, and show that for any $k \geq 1$, two ladders are compatible if and only if the corresponding tableaux satisfy the criterion. This generalizes Leclerc and Zelevinsky's result that two Pl\"{u}cker coordinates are compatible if and only if they are weakly separated.