Cluster Automorphisms

TL;DR

This paper introduces the concept of cluster automorphisms in cluster algebras, analyzing their structure and explicitly computing the automorphism groups for various types including Dynkin and Euclidean.

Contribution

It defines cluster automorphisms and provides detailed analysis and explicit computations of their groups for specific classes of cluster algebras.

Findings

Cluster automorphisms are $ ext{ZZ}$-automorphisms that permute clusters and commute with mutations.

Explicit automorphism groups are computed for Dynkin and Euclidean types.

The structure of automorphism groups is characterized for acyclic and surface-based cluster algebras.

Abstract

In this article, we introduce the notion of cluster automorphism of a given cluster algebra as a $\ZZ$-automorphism of the cluster algebra that sends a cluster to another and commutes with mutations. We study the group of cluster automorphisms in detail for acyclic cluster algebras and cluster algebras from surfaces, and we compute this group explicitly for the Dynkin types and the Euclidean types.