A conjecture on cluster automorphisms of cluster algebras

TL;DR

This paper proves a conjecture that all cluster automorphisms of a cluster algebra are precisely those automorphisms that send clusters to clusters, confirming a key structural property.

Contribution

The paper establishes that every cluster automorphism is a $ ext{Z}$-algebra homomorphism that maps clusters to clusters, resolving a conjecture in the theory of cluster algebras.

Findings

Proves that cluster automorphisms are exactly cluster-preserving algebra automorphisms.

Confirms the conjecture by Chang and Schiffler.

Strengthens understanding of the symmetry structure of cluster algebras.

Abstract

A cluster automorphism is a $\mathbb{Z}$-algebra automorphism of a cluster algebra $\mathcal A$ satisfying that it sends a cluster to another and commutes with mutations. Chang and Schiffler conjectured that a cluster automorphism of $\mathcal A$ is just a $\mathbb{Z}$-algebra homomorphism of a cluster algebra sending a cluster to another. The aim of this article is to prove this conjecture.