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 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. 12 3.1. Flips and the (I, J)-Tamari lattice 132010 Mathematics Subject Classification. 05E45, 05E10, 52B22.

The ν-Tamari Lattice via ν-Trees, ν-Bracket Vectors, and Subword Complexes

“…Concerning the latter, the definition of the multiassociahedron can be naturally generalized to ν-trees, giving rise to the (k, ν)-Tamari complex, which is also a subword complex (these are the complexes of (k + 1)-diagonal free subsets and k-north-east fillings considered in [24] and [42]). For special choices of k and ν, we show that the facet adjacency graph of the (k, ν)-Tamari complex can be realized as the edge graph of a polytopal subdivision of a multiassociahedron (Proposition 6.1), partially extending previous results for the case k = 1 [12]. It would be interesting to know whether a similar result might hold for more general k and ν (Question 6.2).…”

“…It has been recently introduced by Préville-Ratelle and Viennot [37] as a further generalization of the m-Tamari lattice on Fuss-Catalan paths, which was first considered by F. Bergeron and Préville-Ratelle in connection to the combinatorics of higher diagonal coinvariant spaces [4]. These lattices have attracted considerable attention in other areas such as representation theory and Hopf algebras [9,11,32], and remarkable enumerative, algebraic, combinatorial and geometric properties have been discovered [3,10,12,14,19].…”

“…The set of ν-trees is naturally equipped with the simplicial complex structure of the corresponding subword complex, where covering pairs of ν-trees represent adjacent facets. This generalization of the simplicial associahedron was already considered in the context of the ν-Tamari lattice in[12].5.2. Rubey's lattice conjecture.…”

“…In[12], the concept of (I, J)-trees was introduced in order to produce geometric realizations of ν-Tamari lattices. The ν-trees presented in this paper are equivalent to the grid representation of the (I, J)-trees (cf [12,. Remark 2.2]).…”

Cambrian triangulations and their tropical realizations

“…In the classical Tamari case when ε = − n , the function in the proof of Proposition 32 can be replaced by In this section, we exploit the triangulation T ε to obtain a geometric realization of the (ε, I • , J • )lattice as the edge graph of a polyhedral complex induced by a tropical hyperplane arrangement. We follow the same lines as [CPS18], relying on work of M. Develin and B. Sturmfels [DS04]. Define the following geometric objects in the tropical projective space TP |J•|−1 = R J• /R1 1:…”