Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions
| Autor: | Alexander Garver, Thomas McConville |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Mathematics::Combinatorics
Geodesic business.industry 010102 general mathematics Quiver 0102 computer and information sciences 01 natural sciences Theoretical Computer Science Combinatorics Simplicial complex Shard Computational Theory and Mathematics 010201 computation theory & mathematics Lattice (order) Discrete Mathematics and Combinatorics 0101 mathematics Tamari lattice business Quotient Mathematics Subdivision |
| Zdroj: | Journal of Combinatorial Theory, Series A. 158:126-175 |
| ISSN: | 0097-3165 |
| DOI: | 10.1016/j.jcta.2018.03.014 |
| Popis: | Given a tree embedded in a disk, we introduce a simplicial complex of noncrossing geodesics supported by the tree, which we call the noncrossing complex. The facets of the noncrossing complex have the structure of an oriented flip graph. Special cases of these oriented flip graphs include the Tamari lattice, type A Cambrian lattices, Stokes posets of quadrangulations, and oriented exchange graphs of quivers mutation-equivalent to a type A Dynkin quiver. We prove that the oriented flip graph is a polygonal, congruence-uniform lattice. To do so, we express the oriented flip graph as a lattice quotient of a lattice of biclosed sets. The facets of the noncrossing complex have an alternate ordering known as the shard intersection order. We prove that this shard intersection order is isomorphic to a lattice of noncrossing tree partitions, which generalizes the classical lattice of noncrossing set partitions. The oriented flip graph inherits a cyclic action from its congruence-uniform lattice structure. On noncrossing tree partitions, this cyclic action generalizes the classical Kreweras complementation on noncrossing set partitions. |
| Databáze: | OpenAIRE |
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