Cluster characters II: A multiplication formula

TL;DR

This paper proves a general multiplication formula for cluster characters in certain 2-Calabi-Yau categories, extending previous formulas and applicable to various algebraic structures like cluster and preprojective categories.

Contribution

It introduces a unified multiplication formula for cluster characters in Hom-finite triangulated 2-Calabi-Yau categories with cluster tilting objects, generalizing earlier results.

Findings

General multiplication formula for cluster characters.

Applicable to cluster categories, generalized cluster categories, and stable categories.

Extends previous formulas by Caldero-Keller, Xiao-Xu, and Geiss-Leclerc-Schröer.

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-Schr\"oer in the context of preprojective algebras.