A Multiplication Formula for the Modified Caldero\u2013Chapoton Map

Abstract

A frieze in the modern sense is a map from the set of objects of a triangulated category mathsf {C} to some ring. A frieze X is characterised by the property that if tau xrightarrow yrightarrow x is an Auslander–Reiten triangle in mathsf {C}, then X(tau x)X(x)-X(y)=1. The canonical example of a frieze is the (original) Caldero–Chapoton map, which send objects of cluster categories to elements of cluster algebras. Holm and Jørgensen (Nagoya Math J 218:101–124, 2015; Bull Sci Math 140:112–131, 2016), the notion of generalised friezes is introduced. A generalised frieze X' has the more general property that X'(tau x)X'(x)-X'(y)in {0,1}. The canonical example of a generalised frieze is the modified Caldero–Chapoton map, also introduced in Holm and Jørgensen (2015, 2016). Here, we develop and add to the results in Holm and Jørgensen (2016). We define Condition F for two maps alpha and beta in the modified Caldero–Chapoton map, and in the case when mathsf {C} is 2-Calabi–Yau, we show that it is sufficient to replace a more technical “frieze-like” condition from Holm and Jørgensen (2016). We also prove a multiplication formula for the modified Caldero–Chapoton map, which significantly simplifies its computation in practice.