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 $\epsilon \in \{\pm {\}}^n$, we consider a family of $\epsilon$-trees in bijection with the triangulations of the $\epsilon$-polygon. These $\epsilon$-trees define a flag regular triangulation ${\mathcal{T}}^{\epsilon }$ of the subpolytope $\mathrm{conv}\{({\mathbf{e}}_{i_{\bullet }},{\mathbf{e}}_{j_{\circ }})|0\le i_{\bullet }<j_{\circ }\le n+1\}$ of the product of simplices ${△}_{\{0_{\bullet },\dots ,n_{\bullet }\}}\times {△}_{\{1_{\circ },\dots ,(n+1{)}_{\circ }\}}$. The oriented dual graph of the triangulation ${\mathcal{T}}^{\epsilon }$ is the Hasse diagram of the (type $A$) $\epsilon$-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 ${\mathcal{T}}_{I_{\bullet },J_{\circ }}^{\epsilon }$ of the triangulation ${\mathcal{T}}^{\epsilon }$ to the face ${△}_{I_{\bullet }}\times {△}_{J_{\circ }}$. Its dual graph is naturally interpreted as the increasing flip graph on certain $(\epsilon ,I_{\bullet },J_{\circ })$-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 ${\mathcal{T}}_{I_{\bullet },J_{\circ }}^{\epsilon }$ as a polyhedral complex induced by a tropical hyperplane arrangement.

16 pages, 11 figures

References in corpus (2)

• Cambrian Hopf Algebras
• The enumeration of generalized Tamari intervals