2012/04/23 by Christophe Reutenauer, Reutenauer, Christophe
Mathematics · #11B83 #13F60 #68Q70 #68R15 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.NT #msc:11B83 #msc:13F60 #msc:68Q70 #msc:68R15
paper · pdf · doi:10.48550/arxiv.1204.5145
37 pages
arxiv created 2012/04/23 · openalex publication_date 2012/04/23 · arxiv updated 2012/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by a construction in the theory of cluster algebras (Fomin and Zelevinsky), one associates to each acyclic directed graph a family of sequences of natural integers, one for each vertex; this construction is called a \em frieze; these sequences are given by nonlinear recursions (with division), and the fact that they are integers is a consequence of the Laurent phenomenon of Fomin and Zelevinsky. If the sequences satisfy a linear recursion with constant coefficients, then the graph must be a Dynkin diagram or an extended Dynkin diagram, with an acyclic orientation. The converse also holds: the sequences of the frieze associated to an oriented Dynkin or Euclidean diagram satisfy linear recursions, and are even \mathbb N-rational. One uses in the proof objects called SL2-\em tilings of the plane, which are fillings of the discrete plane such that each adjacent 2 by 2 minor is equal to 1. These objects, which have applications in the theory of cluster algebras, are interesting for themselves. Some problems, conjectures and exercises are given.