On denominator conjecture for cluster algebras of finite type

Abstract

We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type. The new contribution is a proof of this conjecture for cluster algebras of type $\mathbb{D}$ and an algorithm for the exceptional types. For the type $\mathbb{D}$ cases, our approach involves geometric model provided by discs with a puncture. By removing the puncture or changing the puncture to an unmarked boundary component, this also yields an alternative proof for the denominator conjecture of cluster algebras of type $\mathbb{A}$ and $\mathbb{C}$ respectively.