Scattering diagrams, tight gradings, and generalized positivity

TL;DR

This paper introduces tight gradings to provide a new, positive, and elementary formula for rank-2 scattering diagrams, leading to proofs of positivity properties in generalized cluster algebras.

Contribution

It defines tight gradings and uses them to derive a positive, explicit formula for rank-2 scattering diagrams, advancing understanding of positivity in generalized cluster algebras.

Findings

Coefficients of wall-functions are positive in any rank generalized cluster scattering diagram.

The formula for rank-2 scattering diagrams is directly computable and manifestly positive.

Results imply Laurent positivity and strong positivity of theta bases in generalized cluster algebras.

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 discover a new 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.