Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Burcroff, Amanda, Lee, Kyungyong, Mou, Lang
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415337366
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