Monoidal categorification from alternating snakes

TL;DR

This paper introduces a new categorical framework based on alternating snakes, establishing a monoidal categorification of a cluster algebra of type A_N through finite dimensional modules of quantum affine algebras.

Contribution

It proves that alternating snakes define a canonical monoidal category with finitely many prime objects, linking it to cluster algebra categorification.

Findings

The category has finitely many prime objects.

Grothendieck ring is isomorphic to that of a known category.

Provides a monoidal categorification of a type A_N cluster algebra.

Abstract

In a recent paper, the authors introduced the notion of an alternating snake and a corresponding family of finite dimensional modules for the quantum affine algebra associated to $A_n$. We prove that under some restrictions, an alternating snake defines a canonical monoidal category. We prove that this category has finitely many prime objects. As a consequence we prove that the Grothendieck ring is isomorphic to the Grothendieck ring of the category $\mathscr C_\xi$ for a suitable height function. In particular it follows that the special family of alternating snakes provides a monoidal categorification of a cluster algebra of type $A_N$ for a suitable value of $N$.