Cluster characters for triangulated 2-Calabi--Yau categories

TL;DR

This paper defines a new cluster character for 2-Calabi--Yau categories, establishing a bijection between indecomposable rigid objects and cluster variables, confirming a conjecture in the finite and acyclic cases.

Contribution

It introduces a cluster character for arbitrary cluster-tilting objects in 2-Calabi--Yau categories and proves a conjecture relating rigid objects to cluster variables.

Findings

The map X(T,L) is a cluster character satisfying a multiplication formula.

Bijection between indecomposable rigid objects and cluster variables in finite and acyclic cases.

Confirms a conjecture of Caldero and Keller.

Abstract

Starting from an arbitrary cluster-tilting object $T$ in a 2-Calabi--Yau category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object $L$, a fraction $X(T,L)$ using a formula proposed by Caldero and Keller. We show that the map taking $L$ to $X(T,L)$ is a cluster character, i.e. that it satisfies a certain multiplication formula. We deduce that it induces a bijection, in the finite and the acyclic case, between the indecomposable rigid objects of the cluster category and the cluster variables, which confirms a conjecture of Caldero and Keller.