Cluster characters II: A multiplication formula
Résumé
Let $\mathcal{C}$ be a Hom-finite triangulated 2-Calabi--Yau category with a cluster tilting object. Under some constructibility assumptions on $\mathcal{C}$ which are satisfied for instance by cluster categories, by generalized cluster categories and by stable categories of modules over a preprojective algebra, we prove a multiplication formula for the cluster character associated with any cluster tilting object. This formula generalizes those obtained by Caldero--Keller for representation finite path algebras and by Xiao--Xu for finite-dimensional path algebras. It is analogous to a formula obtained by Geiss--Leclerc--Schröer in the context of preprojective algebras.
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