Séminaire Lotharingien de Combinatoire, 80B.75 (2018), 12 pp.
Emily Barnard
The Canonical Join Complex of the Tamari Lattice
Abstract.
In this paper, we study a simplicial complex on the elements of the
Tamari lattice in types A and B called the canonical join complex.
The canonical join representation of an element w in a lattice L
is the unique lowest expression VA for w.
We abuse notation and also say that the set A is a canonical join
representation (when we mean VA is a canonical join
representation).
The collection of all such subsets is an abstract simplicial complex
called the canonical join complex of L.
We realize the canonical join complex of the Tamari lattice as a
complex of noncrossing arc diagrams, give a shelling order on its
facets, and show that it is homotopy equivalent to a wedge of
Catalan-many spheres.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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