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

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.