A note on cluster automorphism groups

TL;DR

This paper proposes a new characterization of cluster automorphisms as algebra homomorphisms that bijectively map clusters, relaxing previous mutation-commutation requirements, and proves this in bipartite cases.

Contribution

It introduces a novel algebraic characterization of cluster automorphisms and proves it for bipartite cluster cases, expanding understanding of their structure.

Findings

Conjecture on cluster automorphisms as algebra homomorphisms

Proof of the conjecture for bipartite clusters

Relaxation of mutation-commutation condition

Abstract

We conjecture a characterization of a cluster automorphism as an algebra homomorphism from the cluster algebra to itself that restricts to a bijection between two clusters. This formulation does not require that the map commutes with mutations as in the original definition of cluster automorphisms. We prove the conjecture in the case where at least one of the two clusters is bipartite.