On denominator conjecture for cluster algebras of finite type
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916484909891584 |
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| author | Fu, Changjian Geng, Shengfei |
| author_facet | Fu, Changjian Geng, Shengfei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10914 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On denominator conjecture for cluster algebras of finite type Fu, Changjian Geng, Shengfei Representation Theory Combinatorics Rings and Algebras 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. |
| title | On denominator conjecture for cluster algebras of finite type |
| topic | Representation Theory Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2409.10914 |