Cluster characters II: a multiplication formula

Let C be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object. Under some constructibility assumptions on C which are satisfied, for instance, by cluster categories, by generalized cluster categories and by stable categories of modules over a preprojective algebra of Dynkin type, 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. We prove an analogous formula for the cluster character defined by Fu-Keller in the set-up of Frobenius categories. It is similar to a formula obtained by Geiss-Leclerc-Schroer in the context of preprojective algebras.