Scattering diagrams, tight gradings, and generalized positivity

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Abstract
In 2013, Lee, Li, and Zelevinsky introduced combinatorial objects called compatible pairs to construct the greedy bases for rank-2 cluster algebras, consisting of indecomposable positive elements including the cluster monomials. Subsequently, Rupel extended this construction to the setting of generalized rank-2 cluster algebras by defining compatible gradings. We find a class of combinatorial objects which we call tight gradings. Using this, we give a directly computable, manifestly positive, and elementary but highly nontrivial formula describing rank-2 consistent scattering diagrams. This allows us to show that the coefficients of the wall-functions on a generalized cluster scattering diagram of any rank are positive, which implies the Laurent positivity for generalized cluster algebras and the strong positivity of their theta bases.
Funding Information
  • National Science Foundation (DMS 2042786)
  • National Science Foundation (DMS 2302620)
  • Deutsche Forschungsgemeinschaft (Project-ID 281071066 -- TRR 191)
  • NSF | National Science Foundation Graduate Research Fellowship Program (DGE 2140743)
  • Jack Kent Cooke Foundation (N/A)

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