A Caldero-Chapoton map depending on a torsion class

TL;DR

This paper extends the Caldero-Chapoton map to cases where the object R corresponds to a general Ptolemy diagram, broadening the connection between cluster categories and frieze patterns beyond triangulations and angulations.

Contribution

It generalizes the Caldero-Chapoton map to include cases where R corresponds to a Ptolemy diagram, representing the most general torsion class.

Findings

Extended the Caldero-Chapoton map to Ptolemy diagrams

Connected cluster categories with general frieze patterns

Broadened the class of objects related to frieze patterns

Abstract

Frieze patterns of integers were studied by Conway and Coxeter. Let $\mathscr{C}$ be the cluster category of Dynkin type $A_n$. Indecomposables in $\mathscr{C}$ correspond to diagonals in an $(n+3)$-gon. Work done by Caldero and Chapoton showed that the Caldero-Chapoton map (which is a map dependent on a fixed object $R$ of a category, and which goes from the set of objects of that category to $\mathbb{Z}$), when applied to the objects of $\mathscr{C}$ can recover these friezes. This happens precisely when $R$ corresponds to a triangulation of the $(n+3)$-gon. Later work by authors such as Bessenrodt, Holm, Jorgensen and Rubey generalised this connection with friezes further, now to $d$-angulations of the $(n+3)$-gon with $R$ basic and rigid. In this paper, we extend these generalisations further still, to the case where the object $R$ corresponds to a general Ptolemy diagram, i.e. $R$ is basic and $\textrm{add}(R)$ is the most general possible torsion class.