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arXiv · 2409.10914

On denominator conjecture for cluster algebras of finite type

Changjian Fu·Shengfei Geng
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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.

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Changjian Fu, Shengfei Geng. 2024-09-17. On denominator conjecture for cluster algebras of finite type. https://arxiv.org/abs/2409.10914

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