On denominator conjecture for cluster algebras of finite type

TL;DR

This paper proves the denominator conjecture for finite type cluster algebras, including new proofs for types D, A, and C, using geometric models and algorithms for exceptional types.

Contribution

It confirms the denominator conjecture for all finite type cluster algebras, providing a new proof for type D and algorithms for exceptional types.

Findings

Proved the conjecture for type D cluster algebras.

Provided an algorithm for exceptional types.

Offered alternative proofs for types A and C using geometric models.

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.