CAMBRIAN TRIANGULATIONS AND THEIR TROPICAL REALIZATIONS VINCENT PILAUD Abstract. This paper develops a Cambrian extension of the work of C. Ceballos, A. Padrol and C. Sarmiento on ν-Tamari lattices and their tropical realizations. For any signature " 2 {±}n, we consider a family of "-trees in bijection with the triangulations of the "-polygon. These "-trees " define a flag regular triangulation T of the subpolytope conv f(ei• ; ej◦ ) j 0 ≤ i• < j◦ ≤ n + 1g of the product of simplices 4f0•;:::;n•g × 4f1◦;:::;(n+1)◦g. The oriented dual graph of the triangulation T " is the Hasse diagram of the (type A) "-Cambrian lattice of N. Reading. For " any I• ⊆ f0•; : : : ; n•g and J◦ ⊆ f1◦;:::; (n + 1)◦g, we consider the restriction T of the I•;J◦ " triangulation T to the face 4I• ×4J◦ . Its dual graph is naturally interpreted as the increasing flip graph on certain ("; I•;J◦)-trees, which is shown to be a lattice generalizing in particular the ν-Tamari lattices in the Cambrian setting. Finally, we present an alternative geometric realization of T " as a polyhedral complex induced by a tropical hyperplane arrangement. I•;J◦ The Tamari lattice is a fundamental structure on Catalan objects (such as triangulations, binary trees, or Dyck paths). Introduced by D. Tamari in [Tam51], it has been extensively studied and extended in several directions, see in particular [MHPS12] and the references therein. Our objective is to explore the connection between two generalizations of the Tamari lattice: the (type A) Cambrian lattices of N. Reading [Rea06] and the ν-Tamari lattices of L.-F. Préville-Ratelle and X. Viennot [PRV17]. These two generalizations have strong algebraic roots, in connection to cluster algebras [FZ02, FZ03] and multivariate diagonal harmonics [Ber13, BPR12]. This paper heavily relies on the work of C. Ceballos, A. Padrol and C. Sarmiento on the ge- ometry of ν-Tamari lattices [CPS18]. They start from a family of non-crossing alternating trees in bijection with the triangulations of the (n + 2)-gon. These trees define a flag regular trian- : gulation T of the subpolytope U = conv f(ei• ; ej◦ ) j 0 ≤ i• < j◦ ≤ n + 1g of the product of sim- plices 4f0•;:::;n•g × 4f1◦;:::;(n+1)◦g. The dual graph of this triangulation T is the Hasse diagram of the Tamari lattice. Note that this interpretation of the Tamari lattice as the dual graph of the non-crossing triangulation is ubiquitous in the literature as discussed in [CPS18, Sect. 1.4]. For any subsets I ⊆ f0 ; : : : ; n g and J ⊆ f1 ;:::; (n + 1) g, they consider the restriction T of the • • • ◦ ◦ ◦ I•;J◦ triangulation T to the face 4 × 4 . The simplices of T correspond to certain non-crossing I• J◦ I•;J◦ alternating (I•;J◦)-trees which are in bijection with Dyck paths above a fixed path ν(I•;J◦). More- over, the dual graph of T is the flip graph on (I ;J )-trees, isomorphic to the ν(I ;J )-Tamari I•;J◦ • ◦ • ◦ poset of [PRV17]. This poset actually embeds as an interval of the classical Tamari lattice and is therefore itself a lattice. This interpretation provides three geometric realizations of the ν(I•;J◦)- Tamari lattice [CPS18, Thm. 1.1]: as the dual of the regular triangulation T , as the dual of a I•;J◦ coherent mixed subdivision of a generalized permutahedron, and as the edge graph of a polyhedral complex induced by a tropical hyperplane arrangement. arXiv:1809.00719v1 [math.CO] 3 Sep 2018 Our objective is to extend this approach in the type A Cambrian setting. For any signa- ture " 2 {±}n, we consider a family of "-trees in bijection with the triangulations of the "-polygon. These "-trees define a flag regular triangulation T " of U whose dual graph is the Hasse diagram of the (type A) "-Cambrian lattice of N. Reading [Rea06]. In contrast to the classical Tamari case (obtained when " = −n), we are not aware that this triangulation of U was considered earlier in the literature and the proof of its regularity is a little more subtle in the Cambrian case. For any I ⊆ f0 ; : : : ; n g and J ⊆ f1 ;:::; (n + 1) g, we then consider the restriction T " of the • • • ◦ ◦ ◦ I•;J◦ " triangulation T to the face 4I• × 4J◦ . Its simplices correspond to certain ("; I•;J◦)-trees and its dual graph is the increasing flip graph on these ("; I•;J◦)-trees. Our main combinatorial result is that this increasing flip graph is still an interval of the "-Cambrian lattice in general. The proof is however more involved than in the classical case (" = −n) since this interval does not anymore correspond to a descent class in general. Finally, we mimic the method of [CPS18, Sect. 5] to obtain an alternative geometric realization of T " as a polyhedral complex induced by a tropical I•;J◦ hyperplane arrangement. Partially supported by the French ANR grants SC3A (15 CE40 0004 01) and CAPPS (17 CE40 0018). 1 2 VINCENT PILAUD 1. ("; I•;J◦)-trees and the ("; I•;J◦)-complex This section defines two polygons and certain families of trees associated to a signature " 2 {±}n. 1.1. Two "-polygons. We consider three decorated copies of the natural numbers: the squares N, the blacks N• and the whites N◦. For n 2 N, we use the standard notation [n] := f1; : : : ; ng and : : : define [[n] = f0; : : : ; ng, [n]] = f1; : : : ; n + 1g and [[n]] = f0; : : : ; n+1g. We write [n], [n•], [n◦] and so on for the decorated versions of these intervals. Fix a signature " 2 {±}n. We consider two convex polygons associated to the signature " as follows: • a (n+2)-gon P" with square vertices labeled by [[n ]] from left to right and where vertex i is above the segment (0; (n + 1)) if "i = + and below it if "i = −. • a (2n + 2)-gon P" with black or white vertices, obtained from P" by replacing the square •−◦ vertex 0 (resp. (n+1)) by the black vertex 0• (resp. white vertex (n+1)◦), and splitting each other square vertex i into a pair of white and black vertices i◦ and i• (such that the " vertices of P•−◦ are alternatively colored black and white). The black (resp. white) vertices " of P•−◦ are labeled by [[n•] (resp. [n◦]]) from left to right. Examples of these polygons are represented in Figure1 for the signature " = −++−+−−+. 3 5 3 5◦5 2 2 2• 3◦ • • ◦ 8 8 ◦8• 0 9 0• 9◦ 1 1◦ 1• 7 7◦7• 4 6 4◦4• 6◦6• The polygons P" (left) and P" (right) for the signature " = −++−+−−+. Figure 1. •−◦ 1.2. ("; I•;J◦)-trees. All throughout the paper, we consider I• ⊆ [[n•] and J◦ ⊆ [n◦]] and we always assume that min(I ) < min(J ) and max(I ) < max(J ). Consider the graph G" with • ◦ • ◦ I•;J◦ vertices I• [ J◦ and edges f(i•; j◦) j i• 2 I•; j◦ 2 J◦; i• < j◦g. Note that this graph is geometric: " " its vertices are considered as vertices of P•−◦ and its edges are considered as straight edges in P•−◦. A subgraph of G" is non-crossing if no two of its edges cross in their interior. I•;J◦ Proposition 1. Any maximal non-crossing subgraph of G" is a spanning tree of G" . I•;J◦ I•;J◦ Proof. The proof works by induction on jI•j + jJ◦j. The result is immediate when jI•j = jJ◦j = 1. Assume now for instance that jI•j > 1 (the case jI•j = 1 and jJ◦j > 1 is similar). Let i• := max(I•) and j◦ := min fj◦ 2 J◦ j i• < j◦ and "i = "jg [ fmax(J◦)g . Note that our choice of j◦ ensures that (i ; j ) is a boundary edge of conv(I [ J ). Moreover, any edge of G" incident to i is of • ◦ • ◦ I•;J◦ • the form (i ; j0 ) for i < j0 while any edge of G" incident to j is of the form (i0 ; j ) for i0 ≤ i • ◦ • ◦ I•;J◦ ◦ • ◦ • • (by maximality of i ). Therefore, all edges of G" f(i ; j )g incident to i cross all edges • I•;J◦ r • ◦ • of G" f(i ; j )g incident to j . Consider now a maximal non-crossing subgraph t of G" . I•;J◦ r • ◦ ◦ I•;J◦ Then t contains the edge (i•; j◦) (since t is maximal) and either i• or j◦ is a leaf in t (since t is 0 non-crossing). Assume for example that i• is a leaf and let I• := I• r fi•g. Then t r f(i•; j◦)g is a " maximal non-crossing subgraph of G 0 (the maximality is ensured from the fact that (i•; j◦) is I•;J◦ " a boundary edge of conv(I• [ J◦)). By induction, t f(i•; j◦)g is thus a spanning tree of G 0 , r I•;J◦ so that t is a spanning tree of G" . I•;J◦ In accordance to Proposition1, we define a ("; I•;J◦)-forest to be a non-crossing subgraph of G" , and a ("; I ;J )-tree to be a maximal ("; I ;J )-forest. Note that a ("; I ;J )-tree I•;J◦ • ◦ • ◦ • ◦ has jI•j + jJ◦j − 1 edges. Examples can be found in Figure2. CAMBRIAN TRIANGULATIONS AND THEIR TROPICAL REALIZATIONS 3 3◦3• 5◦5• 3◦3• 5◦5• 2◦ 2◦ 8• 8• 0• 9◦ 0• 9◦ 1◦ 1◦ 1• 1• 4• 6◦6• 4• 6◦6• Figure 2. A ("; I•;J◦)-forest (left) and a ("; I•;J◦)-tree (right) for " = −++−+−−+, I• = [[8•] r f2•; 7•g and J◦ = [8◦]] r f4◦; 7◦; 8◦g. 1.3. The ("; I ;J )-complex. We call ("; I ;J )-complex C" the clique complex of the graph • ◦ • ◦ I•;J◦ of non-crossing edges of G" .