A combinatorial approach to scattering diagrams
arXiv:1806.05094 · doi:10.5802/alco.107
Abstract
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. In acyclic finite type, we construct cluster scattering diagrams of acyclic finite type from Cambrian fans and sortable elements, with a simple direct proof.
The second half of arXiv:1712.06968, which was originally "Scattering diagrams and scattering fans". The contents of this paper will be removed from arXiv:1712.06968, which will be re-titled "Scattering fans." Version 2: Minor expository changes. (We thank some anonymous referees for helpful comments.) Version 3: Corrections to reconcile the arXiv version with the final pre-publication version
References in corpus (11)
- Sortable elements and Cambrian lattices
- Lattice congruences of the weak order
- Universal geometric cluster algebras
- Sortable elements in infinite Coxeter groups
- Cluster algebras and continued fractions
- The greedy basis equals the theta basis
- Combinatorial frameworks for cluster algebras
- The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
- Scattering fans
- The Shard Intersection Order on Permutations
- Stable Cluster Variables
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- Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams
- Dominance phenomena: mutation, scattering and cluster algebras