Oriented Flip Graphs and Noncrossing Tree Partitions

Alexander Garver,Thomas Mcconville

doi:10.46298/dmtcs.6379

Abstract

Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.

Highlights

The facets of a pure, thin simplicial complex form a graph, often called a flip graph, where two facets are adjacent if their intersection is a face of codimension 1
We define the flip graph of T, denoted F G(T ), to be the graph whose vertices are facets of ∆NC(T ) and two vertices are connected by an edge if the corresponding facets differ by a single flip
A lattice L is congruence-uniform if it may be constructed from a 1-element lattice by a sequence of doublings

Summary

Introduction

The facets of a pure, thin simplicial complex form a graph, often called a flip graph, where two facets are adjacent if their intersection is a face of codimension 1. Using the theory of cluster algebras, one may define many different orientations on the graph of tridanengoutleadtio−F−n→Gs o(Tf a polygon. To prove Theorem 1.1, map from biclosed sets to twrieandgeufilnaetioannsatuhxaitliiadreyntliafitteisce−F−o→Gf (bTic)loasseadqsueottsieinnt§l3a.ttTicheeonfwtheedlaetfitinceeaofsubrijcelcotsievde sets Another significant class of Catalan objects are noncrossing partitions [9]. They have generalizations to finite Coxeter groups [1] as well as connections to the theory of cluster algebras [12] and the representation theory of finite dimensional algebras [8]. Oorf dAneosrnacΨrcoo(s−Fn−s→giGnrug(eTtnr)ec)ee,-pwuanhritificothiromnwsleaotrfteiTccea,oll−Fr−di→nGer(§eT4d.)bayWdrmeefiiptnsreoamveaelnttehtreinsafaotelllaootrwtdicienerg.insgukrpnroiwsinngascothneneshctaiordn intersection between the oriented flip graph and the lattice of noncrossing tree partitions. Theorem 1.2 into the language of cluster algebras, we give a nice characterization of c-matrices associated to Q Proofs will be given in the full version of the paper

Noncrossing complex

Lattice quotient description of oriented flip graphs

Congruence-uniformity

Noncrossing tree partitions

Connection to cluster algebras