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June 2015 Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object
Thorsten Holm, Peter Jørgensen
Nagoya Math. J. 218: 101-124 (June 2015). DOI: 10.1215/00277630-2891495


The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi–Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.


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Thorsten Holm. Peter Jørgensen. "Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object." Nagoya Math. J. 218 101 - 124, June 2015.


Published: June 2015
First available in Project Euclid: 11 May 2015

zbMATH: 1314.05221
MathSciNet: MR3345625
Digital Object Identifier: 10.1215/00277630-2891495

Primary: 05E10
Secondary: 13F60 , 16G70 , 18E30

Keywords: Auslander–Reiten triangle , cluster category , polygon dissection , rigid subcategory , Serre functor , triangulated category

Rights: Copyright © 2015 Editorial Board, Nagoya Mathematical Journal

Vol.218 • June 2015
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