Cluster Monomials are Dual Canonical

TL;DR

This paper extends the proof that cluster monomials are in the dual canonical basis from symmetric types to all Lie types by employing a folding technique in the context of monoidal categorification.

Contribution

It generalizes the result that cluster monomials are in the dual canonical basis to all Lie types using a folding approach.

Findings

Cluster monomials are in the dual canonical basis for all Lie types.

Folding technique effectively generalizes previous symmetric type results.

Monoidal categorification via KLR algebra representations is central to the proof.

Abstract

Kang, Kashiwara, Kim and Oh have proved that cluster monomials lie in the dual canonical basis, under a symmetric type assumption. This involves constructing a monoidal categorification of a quantum cluster algebra using representations of KLR algebras. We use a folding technique to generalise their results to all Lie types.