Cluster automorphisms and compatibility of cluster variables

TL;DR

This paper introduces unistructural cluster algebras, proves their properties for certain types, and explores conditions for cluster automorphisms, also addressing a key conjecture on variable compatibility.

Contribution

It defines unistructural cluster algebras, proves their properties for Dynkin and rank 2 types, and characterizes cluster automorphisms in these contexts.

Findings

Cluster algebras of Dynkin type are unistructural.

Cluster algebras of rank 2 are unistructural.

Automorphisms of unistructural or Euclidean type are characterized by permutations of variables.

Abstract

In this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters. We prove that cluster algebras of Dynkin type and cluster algebras of rank 2 are unistructural, then prove that if $\mathcal{A}$ is unistructural or of Euclidean type, then $f: \mathcal{A}\to \mathcal{A}$ is a cluster automorphism if and only if $f$ is an automorphism of the ambient field which restricts to a permutation of the cluster variables. In order to prove this result, we also investigate the Fomin-Zelevinsky conjecture that two cluster variables are compatible if and only if one does not appear in the denominator of the Laurent expansions of the other.