Motivic cluster multiplication formulas in 2-Calabi-Yau categories

TL;DR

This paper develops motivic cluster characters using virtual Poincaré polynomials and proves a motivic multiplication formula for weighted quantum cluster characters in 2-Calabi-Yau categories, extending previous results.

Contribution

It introduces motivic cluster characters and establishes a motivic multiplication formula for quantum cluster characters in 2-Calabi-Yau categories, including a refined version.

Findings

Proved a motivic version of multiplication formulas for quantum cluster characters.

Introduced motivic cluster characters via virtual Poincaré polynomials.

Extended formulas to the case of cluster categories of acyclic quivers.

Abstract

We introduce a notion of motivic cluster characters via virtual Poincar\'{e} polynomials, and prove a motivic version of multiplication formulas obtained by Chen-Xiao-Xu for weighted quantum cluster characters associated to a 2-Calabi-Yau triangulated category $\mathcal{C}$ with a cluster tilting object. Furthermore, a refined form of this formula is also given. When $\mathcal{C}$ is the cluster category of an acyclic quiver, our certain refined multiplication formula is a motivic version of the multiplication formula in [International Mathematics Research Notices, rnad172(2023)].