Positivity of generalized cluster scattering diagrams
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912260789633024 |
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| author | Burcroff, Amanda Lee, Kyungyong Mou, Lang |
| author_facet | Burcroff, Amanda Lee, Kyungyong Mou, Lang |
| contents | We introduce a new class of combinatorial objects, named tight gradings, which are certain nonnegative integer-valued functions on maximal Dyck paths. Using tight gradings, we derive a manifestly positive formula for any wall-function in a rank-2 generalized cluster scattering diagram. We further prove that any consistent rank-2 scattering diagram is positive with respect to the coefficients of initial wall-functions. Moreover, our formula yields explicit expressions for relative Gromov-Witten invariants on weighted projective planes and the Euler characteristics of moduli spaces of framed stable representations on complete bipartite quivers. Finally, by leveraging the rank-2 positivity, we show that any higher-rank generalized cluster scattering diagram has positive wall-functions, which leads to a proof of the positivity of the Laurent phenomenon and the strong positivity of Chekhov-Shapiro's generalized cluster algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positivity of generalized cluster scattering diagrams Burcroff, Amanda Lee, Kyungyong Mou, Lang Combinatorics Commutative Algebra Algebraic Geometry Rings and Algebras Representation Theory 13F60, 05E10, 14N35 We introduce a new class of combinatorial objects, named tight gradings, which are certain nonnegative integer-valued functions on maximal Dyck paths. Using tight gradings, we derive a manifestly positive formula for any wall-function in a rank-2 generalized cluster scattering diagram. We further prove that any consistent rank-2 scattering diagram is positive with respect to the coefficients of initial wall-functions. Moreover, our formula yields explicit expressions for relative Gromov-Witten invariants on weighted projective planes and the Euler characteristics of moduli spaces of framed stable representations on complete bipartite quivers. Finally, by leveraging the rank-2 positivity, we show that any higher-rank generalized cluster scattering diagram has positive wall-functions, which leads to a proof of the positivity of the Laurent phenomenon and the strong positivity of Chekhov-Shapiro's generalized cluster algebras. |
| title | Positivity of generalized cluster scattering diagrams |
| topic | Combinatorics Commutative Algebra Algebraic Geometry Rings and Algebras Representation Theory 13F60, 05E10, 14N35 |
| url | https://arxiv.org/abs/2503.03719 |