Cluster automorphisms and compatibility of cluster variables

Ibrahim Assem; Ralf Schiffler; Vasilisa Shramchenko

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arXiv · 1307.4838

Cluster automorphisms and compatibility of cluster variables

Ibrahim Assem·Ralf Schiffler·Vasilisa Shramchenko

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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.

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BibTeX RIS

Ibrahim Assem, Ralf Schiffler, Vasilisa Shramchenko. 2013-07-18. Cluster automorphisms and compatibility of cluster variables. https://arxiv.org/abs/1307.4838

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