Geometry of ν-Tamari lattices in types A and B

In this extended abstract, we exploit the combinatorics and geometry of triangulations of products of simplices to reinterpret and generalize a number of constructions in Catalan combinatorics. 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 of a subpolytope of the cartesian product of two simplices, which provides a geometric realization of the ν-Tamari lattice introduced by Préville-Ratelle and Viennot. Dualizing this triangulation, we obtain a polyhedral complex induced by an arrangement of tropical hyperplanes whose 1-skeleton realizes the Hasse diagram of the ν-Tamari lattice, and thus generalizes the simple associahedron. Specializing to the Fuss-Catalan case realizes the m-Tamari lattices as 1-skeleta of regular subdivisions of classical associahedra, giving a positive answer to a question of F. Bergeron. The simplicial complex underlying our triangulation has its h-vector given by a suitable generalization of the Narayana numbers. We propose it as a natural generalization of the classical simplicial associahedron, alternative to the rational associahedron of Armstrong, Rhoades and Williams. 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.