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| Format: | Preprint |
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2023
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| Online-Zugang: | https://arxiv.org/abs/2310.03306 |
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| _version_ | 1866918143499173888 |
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| author | Geiß, Christof Labardini-Fragoso, Daniel Wilson, Jon |
| author_facet | Geiß, Christof Labardini-Fragoso, Daniel Wilson, Jon |
| contents | We prove that for any possibly-punctured surface with non-empty boundary $\mathbfΣ=(Σ, \mathbb{M}, \mathbb{P})$, and any tagged triangulation $T$ of $\mathbfΣ$ in the sense of Fomin--Shapiro--Thurston, the coefficient-free bangle functions of Musiker--Schiffler--Williams coincide with the coefficient-free generic Caldero--Chapoton functions arising from the Jacobian algebra of the quiver with potential $(Q(T), W(T))$ associated to $T$ by Cerulli Irelli and the second author.
When the set of boundary marked points $\mathbb{M}$ has at least two elements, Schröer and the first two authors have shown, relying heavily on results of Mills, Muller and Qin, that the generic coefficient-free Caldero-Chapoton functions form a basis of the coefficient-free (upper) cluster algebra $\mathcal{A}(\mathbfΣ)=\mathcal{U}(\mathbfΣ)$. So, the set of bangle functions proposed by Musiker--Schiffler--Williams over ten years ago is indeed a basis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_03306 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary Geiß, Christof Labardini-Fragoso, Daniel Wilson, Jon Representation Theory Combinatorics 13F60, 16G20, 05C70 We prove that for any possibly-punctured surface with non-empty boundary $\mathbfΣ=(Σ, \mathbb{M}, \mathbb{P})$, and any tagged triangulation $T$ of $\mathbfΣ$ in the sense of Fomin--Shapiro--Thurston, the coefficient-free bangle functions of Musiker--Schiffler--Williams coincide with the coefficient-free generic Caldero--Chapoton functions arising from the Jacobian algebra of the quiver with potential $(Q(T), W(T))$ associated to $T$ by Cerulli Irelli and the second author. When the set of boundary marked points $\mathbb{M}$ has at least two elements, Schröer and the first two authors have shown, relying heavily on results of Mills, Muller and Qin, that the generic coefficient-free Caldero-Chapoton functions form a basis of the coefficient-free (upper) cluster algebra $\mathcal{A}(\mathbfΣ)=\mathcal{U}(\mathbfΣ)$. So, the set of bangle functions proposed by Musiker--Schiffler--Williams over ten years ago is indeed a basis. |
| title | Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary |
| topic | Representation Theory Combinatorics 13F60, 16G20, 05C70 |
| url | https://arxiv.org/abs/2310.03306 |