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arXiv · 2503.03719

Positivity of generalized cluster scattering diagrams

Amanda Burcroff·Kyungyong Lee·Lang Mou
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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.

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Amanda Burcroff, Kyungyong Lee, Lang Mou. 2025-03-05. Positivity of generalized cluster scattering diagrams. https://arxiv.org/abs/2503.03719

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