A Multiplication Formula for the Modified Caldero Chapoton Map

TL;DR

This paper introduces a multiplication formula for the modified Caldero-Chapoton map, simplifying its computation and extending previous results on generalized friezes in 2-Calabi-Yau categories.

Contribution

It develops a new multiplication formula for the modified Caldero-Chapoton map and introduces Condition F, simplifying the analysis in 2-Calabi-Yau categories.

Findings

Proves a multiplication formula for the modified Caldero-Chapoton map.

Introduces Condition F for maps in the modified Caldero-Chapoton map.

Simplifies computation of the map in 2-Calabi-Yau categories.

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 $\tau x\rightarrow y\rightarrow x$ 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. \par In \cite{friezes1} and \cite{friezes2}, 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 \cite{friezes1} and \cite{friezes2}. \par Here, we develop and add to the results in \cite{friezes2}. We define Condition F for two maps $\alpha$ and $\beta$ in the modified Calero-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 \cite{friezes2}. We also prove a multiplication formula for the modified Caldero-Chapoton map, which significantly simplifies its computation in practice.