Cluster automorphisms and compatibility of cluster variables

Ibrahim Assem (1) , Ralf Schiffler (2) , Vasilisa Shramchenko (1)

1 UdeS - Département de mathématiques [Sherbrooke]
2 UCONN - University of Connecticut

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.

Mots clés: cluster algebra; cluster automorphism; denominators of cluster variables; compatibility of cluster variables

Domaines: Théorie des représentations [math.RT]

Ibrahim Assem, Ralf Schiffler, Vasilisa Shramchenko. Cluster automorphisms and compatibility of cluster variables. 2013. ⟨hal-00845825⟩