Cluster characters II: A multiplication formula

Abstract

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.

v3: Updated references. Section on Fu--Keller's cluster character. v4: Some expository changes as suggested by the referee. To appear in Proceedings LMS

References in corpus (5)

• Calabi-Yau algebras
• Cluster structures for 2-Calabi-Yau categories and unipotent groups
• Cluster algebras via cluster categories with infinite-dimensional morphism spaces
• Cluster algebra structures and semicanonical bases for unipotent groups
• Preprojective algebras and cluster algebras

Cited by in corpus (9)

• Atomic bases in cluster algebras of types A and $\tilde{A}$
• A geometric $q$-character formula for snake modules
• Integral bases of cluster algebras and representations of tame quivers
• Cluster multiplication in regular components via generalized Chebyshev polynomials
• On Generalized Minors and Quiver Representations
• Notes on the cluster multiplication formulas for 2-Calabi-Yau categories
• The cluster multiplication theorem for acyclic quantum cluster algebras
• Extensions in Jacobian algebras via punctured skein relations
• Cluster multiplication theorem in the quantum cluster algebra of type $A_2^{(2)}$