Cambrian triangulations and their tropical realizations

Abstract: This paper develops a Cambrian extension of the work of C. Ceballos, A. Padrol and C. Sarmiento on $\nu$-Tamari lattices and their tropical realizations. For any signature $\varepsilon \in {\pm}^n$, we consider a family of $\varepsilon$-trees in bijection with the triangulations of the $\varepsilon$-polygon. These $\varepsilon$-trees define a flag regular triangulation $\mathcal{T}^\varepsilon$ of the subpolytope $\operatorname{conv} {(\mathbf{e}{i\bullet}, \mathbf{e}{j\circ}) \, | \, 0 \le i_\bullet < j_\circ \le n+1 }$ of the product of simplices $\triangle_{{0_\bullet, \dots, n_\bullet}} \times \triangle_{{1_\circ, \dots, (n+1)\circ}}$. The oriented dual graph of the triangulation $\mathcal{T}^\varepsilon$ is the Hasse diagram of the (type $A$) $\varepsilon$-Cambrian lattice of N. Reading. For any $I\bullet \subseteq {0_\bullet, \dots, n_\bullet}$ and $J_\circ \subseteq {1_\circ, \dots, (n+1)\circ}$, we consider the restriction $\mathcal{T}^\varepsilon{I_\bullet, J_\circ}$ of the triangulation $\mathcal{T}^\varepsilon$ to the face $\triangle_{I_\bullet} \times \triangle_{J_\circ}$. Its dual graph is naturally interpreted as the increasing flip graph on certain $(\varepsilon, I_\bullet, J_\circ)$-trees, which is shown to be a lattice generalizing in particular the $\nu$-Tamari lattices in the Cambrian setting. Finally, we present an alternative geometric realization of $\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ as a polyhedral complex induced by a tropical hyperplane arrangement.