Cluster characters for triangulated 2-Calabi--Yau categories

Details Cluster characters for triangulated 2-Calabi--Yau categories Cluster characters for triangulated 2-Calabi--Yau categories

2007-03-19

arxiv.org/abs/math/0703540v3

Annales de l'Institut Fourier Tome 58, 6 (2008) 2221-2248

Mathematics

Representation Theory

21 pages. The numberings now coincide with those of the published version

Scientific paper

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.

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