A combinatorial approach to scattering diagrams

Understand

Scattering diagrams arose in the context of mirror symmetry, but a special class of scattering diagrams (the cluster scattering diagrams) were recently developed to prove key structural results on cluster algebras.

• We use the connection to cluster algebras to calculate the function attached to the limiting wall of a rank-2 cluster scattering diagram of affine type.
• In the skew-symmetric rank-2 affine case, this recovers a formula due to Reineke.
• In the same case, we show that the generating function for signed Narayana numbers appears in a role analogous to a cluster variable.