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Transactions of the American Mathematical Society

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Geometry of $ \nu$-Tamari lattices in types $ A$ and $ B$


Authors: Cesar Ceballos, Arnau Padrol and Camilo Sarmiento
Journal: Trans. Amer. Math. Soc. 371 (2019), 2575-2622
MSC (2010): Primary 05E45, 05E10, 52B22
DOI: https://doi.org/10.1090/tran/7405
Published electronically: November 27, 2018
MathSciNet review: 3896090
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Abstract: 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 $ \nu $-Tamari lattices. In our framework, the main role of ``Catalan objects'' is played by $ (I,\overline {J})$-trees: bipartite trees associated to a pair $ (I,\overline {J})$ of finite index sets that stand in simple bijection with lattice paths weakly above a lattice path $ \nu =\nu (I,\overline {J})$. Such trees label the maximal simplices of a triangulation whose dual polyhedral complex gives a geometric realization of the $ \nu $-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 $ \nu $-Tamari lattice with a full simplicial complex structure. It is a natural generalization of the classical simplicial associahedron, an 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.


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Additional Information

Cesar Ceballos
Affiliation: Faculty of Mathematics, University of Vienna, Vienna, Austria

Arnau Padrol
Affiliation: Sorbonne Universités, Université Pierre et Marie Curie (Paris 6), Institut de Mathématiques de Jussieu - Paris Rive Gauche (UMR 7586), Paris, France
Address at time of publication: Sorbonne Université, Institut de Mathématiques de Jussieu - Paris Rive Gauche (UMR 7586), Paris, France

Camilo Sarmiento
Affiliation: Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Address at time of publication: Universidad del Norte, Departamento de Matemáticas y Estadística

DOI: https://doi.org/10.1090/tran/7405
Keywords: Triangulations of products of simplices, Tamari lattices, associahedra, cyclohedra, tropical hyperplanes
Received by editor(s): March 9, 2017
Received by editor(s) in revised form: August 31, 2017, and September 12, 2017
Published electronically: November 27, 2018
Additional Notes: The research of the first author was supported by the Austrian Science Foundation FWF, grant F 5008-N15, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”.
The second author was supported by the grant ANR-17-CE40-0018 of the French National Research Agency ANR (project CAPPS) and the program PEPS Jeunes Chercheur-e-s 2016 and 2017 from the INSMI
The third author was partially supported by CDS Magdeburg.
Article copyright: © Copyright 2018 Cesar Ceballos, Arnau Padrol, and Camilo Sarmiento