A Caldero-Chapoton map depending on a torsion class

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.