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

TL;DR

This paper explores the geometry of $ u$-Tamari lattices in types A and B, providing new geometric realizations, generalizations of classical associahedra, and answering open questions about their structure and symmetry.

Contribution

It introduces geometric realizations of $ u$-Tamari lattices via triangulations of simplices, generalizes the associahedron, and extends constructions to type B with new partial orders.

Findings

Geometric realizations of $m$-Tamari lattices as polyhedral subdivisions of associahedra.

A full simplicial complex structure for $ u$-Tamari lattices.

Type B analogues of the constructions and new partial orders.

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 Pr\'evile-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, 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.