Cambrian triangulations and their tropical realizations

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 ε∈{±}ⁿ, we consider a family of ε-trees in bijection with the triangulations of the ε-polygon. These ε-trees define a flag regular triangulation T^ε of the subpolytope conv(e_i• ,e_j∘ )|0≤i_•<j_∘≤n+1 of the product of simplices △_{{0•,…,n•}}×△_{{1∘,…,(n+1)∘}}. The oriented dual graph of the triangulation T^ε is the Hasse diagram of the (type A) ε-Cambrian lattice of N. Reading. For any I_•⊆{0_•,…,n_•} and J_∘⊆{1_∘,…,(n+1)_∘}, we consider the restriction T_I•,J∘ ^ε of the triangulation T^ε to the face △_I• ×△_J∘ . Its dual graph is naturally interpreted as the increasing flip graph on certain (ε,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_I•,J∘ ^ε as a polyhedral complex induced by a tropical hyperplane arrangement.