[Paper Review] Noncrossing hypertrees

Jon McCammond|arXiv (Cornell University)|Jul 20, 2017

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper introduces the noncrossing hypertree complex as a geometric realization of the face poset of a simplicial complex dual to the poset of noncrossing hypertrees, establishing a homeomorphism with the noncrossing partition link—a piecewise spherical complex. The key contribution is a metric and topological structure on the complex, revealing a duality between simplices and spheres labeled by noncrossing hypertrees, and showing that certain apartments are simplicial associahedra, unifying known bijections between noncrossing partitions and associahedra.

ABSTRACT

Hypertrees and noncrossing trees are well-established objects in the combinatorics literature, but the hybrid notion of a noncrossing hypertree has received less attention. In this article I investigate the poset of noncrossing hypertrees as an induced subposet of the hypertree poset. Its dual is the face poset of a simplicial complex, one that can be identified with a generalized cluster complex of type $A$. The first main result is that this noncrossing hypertree complex is homeomorphic to a piecewise spherical complex associated with the noncrossing partition lattice and thus it has a natural metric. The fact that the order complex of the noncrossing partition lattice with its bounding elements removed is homeomorphic to a generalized cluster complex was not previously known or conjectured. The metric noncrossing hypertree complex is a union of unit spheres with a number of remarkable properties: 1) the metric subspheres and simplices in each dimension are both bijectively labeled by the set of noncrossing hypertrees with a fixed number of hyperedges, 2) the number of spheres containing the simplex labeled by the noncrossing tree $τ$ is the same as the number simplices in the sphere labeled by the noncrossing tree $τ$, and 3) among the maximal spherical subcomplexes one finds every normal fan of a metric realization of the simple associahedron associated to the cluster algebra of type $A$. In particular, the poset of noncrossing hypertrees and its metric simplicial complex provide a new perspective on familiar combinatorial objects and a common context in which to view the known bijections between noncrossing partitions and the vertices/facets of simple/simplicial associahedra.

Motivation & Objective

• To define and study the noncrossing hypertree complex as the dual of the face poset of an induced subposet of the hypertree poset.
• To establish a topological and geometric structure on this complex via a homeomorphism to the noncrossing partition link.
• To reveal a duality between simplices and metric spheres in the complex, both labeled by noncrossing hypertrees.
• To show that certain apartments in the complex are isometric to simplicial associahedra, linking the complex to cluster algebras and type A associahedra.
• To provide a new geometric framework for understanding known bijections between noncrossing partitions, noncrossing trees, and associahedra.

Proposed method

• The noncrossing hypertree complex is constructed as the dual of the face poset of the induced subposet of noncrossing hypertrees within the hypertree poset.
• The topology of the complex is identified via a homeomorphism to the noncrossing partition link—the link of the long diagonal edge in the orthoscheme complex of the noncrossing partition lattice.
• A piecewise spherical metric is induced on the complex from the metric of the noncrossing partition link.
• The duality between simplices and spheres is established by showing that each simplex lies in a unique isometric unit sphere, and that the number of spheres containing a simplex labeled by a noncrossing tree τ equals the number of simplices in the sphere labeled by τ.
• The complex is analyzed through the lens of generalized cluster complexes of type A with m=2, identifying it with a full subcomplex of the cluster complex.
• The connection to associahedra is proven by showing that when a tree chamber contains only one partition chamber (i.e., for caterpillar trees), the corresponding apartment is a simplicial associahedron.

Experimental results

Research questions

• RQ1 How can the poset of noncrossing hypertrees be realized as a geometric simplicial complex with a natural metric?
• RQ2 What is the topological structure of the noncrossing hypertree complex, and how does it relate to the noncrossing partition lattice?
• RQ3 Is there a duality between simplices and metric spheres in the noncrossing hypertree complex, and if so, how is it encoded?
• RQ4 Under what conditions does an apartment in the noncrossing hypertree complex become a simplicial associahedron?
• RQ5 Can the known bijections between noncrossing partitions and vertices/facets of associahedra be reinterpreted within this new geometric framework?

Key findings

• The noncrossing hypertree complex is homeomorphic to the noncrossing partition link, a piecewise spherical complex associated with the noncrossing partition lattice.
• The complex inherits a piecewise spherical metric from the noncrossing partition link, giving it a natural geometric structure.
• There is a bijection between the set of top-dimensional simplices (tree chambers) and the set of top-dimensional spherical subcomplexes (apartments), both labeled by noncrossing trees.
• For each noncrossing tree τ, the number of apartments containing the tree chamber labeled τ equals the number of tree chambers contained in the apartment labeled τ.
• When a tree chamber consists of a single partition chamber—specifically for caterpillar trees—the corresponding apartment is a simplicial associahedron.
• The complex contains all normal fans of type A simple associahedra constructed by Hohlweg and Lange, and these correspond to the apartments of caterpillar trees.