2013/07/18 by Ibrahim Assem, Ralf Schiffler, Assem, Ibrahim +3
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1307.4838
13 pages
arxiv created 2013/07/18 · arxiv updated 2013/07/19
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 A is unistructural or of Euclidean type, then f: A→ 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.