Monoidal categorification of cluster algebras

TL;DR

This paper introduces a framework for monoidal categorification of quantum cluster algebras using graded modules over symmetric Khovanov-Lauda-Rouquier algebras, and proves a related conjecture by Leclerc.

Contribution

It defines monoidal categorifications of quantum cluster algebras and provides a mutation criterion, also proving Leclerc's conjecture on basis element products.

Findings

Established a criterion for monoidal categorification via mutations.

Proved Leclerc's conjecture on upper global basis elements.

Demonstrated the applicability to categories of graded modules over R.

Abstract

We give a definition of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification of a quantum cluster algebra, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions once the first-step mutations are possible. In the course of the study, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.