Geometry of ν-Tamari lattices in types A and B
Cesar Ceballos, Arnau Padrol, and Camilo Eduardo Sarmiento Cortés
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Submission date: 29. Nov. 2016
published in: Seminaire lotharingien de combinatoire, 78B (2017), art-no. 68
MSC-Numbers: 05E45, 05E10, 52B45, 52B22
Keywords and phrases: Tamari lattice, Associahedron, Cyclohedron, Triangulations
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In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of ν-Tamari lattices. In our framework, the main role of “Catalan objects” is played by (I,J)-trees: bipartite trees associated to a pair (I,J) of finite index sets that stand in simple bijection with lattice paths weakly above a lattice path ν = ν(I,J). Such trees label the maximal simplices of a triangulation whose dual polyhedral complex gives a geometric realization of the ν-Tamari lattice introduced by Prévile-Ratelle and Viennot. In particular, we obtain geometric realizations of m-Tamari lattices as polyhedral subdivisions of associahedra induced by an arrangement of tropical hyperplanes, giving a positive answer to an open question of F. Bergeron.
The simplicial complex underlying our triangulation endows the ν-Tamari lattice with a full simplicial complex structure. It is a natural generalization of the classical simplicial associahedron, alternative to the rational associahedron of Armstrong, Rhoades and Williams, whose h-vector entries are given by a suitable generalization of the Narayana numbers.
Our methods are amenable to cyclic symmetry, which we use to present type B analogues of our constructions. Notably, we define a partial order that generalizes the type B Tamari lattice, introduced independently by Thomas and Reading, along with corresponding geometric realizations.