Nantel Bergeron, York University  

Title: Interval hypergraphic lattices

Abstract: For a hypergraph~$\mathbb{H}$ on~$[n]$, the hypergraphic poset~$P_{\mathbb{H}}$ is the transitive closure of the oriented skeleton of the hypergraphic polytope~$\triangle_{\mathbb{H}}$ (the Minkowski sum of the standard simplices~$\triangle_H$ for all~$H \in \mathbb{H}$).Hypergraphic posets include the weak order for the permutahedron (when~$\mathbb{H}$ is the complete graph on~$[n]$) and the Tamari lattice for the associahedron (when~$\mathbb{H}$ is the set of all intervals of~$[n]$), and it is natural to study lattice properties of hypergraphic posets.
In this talk, we focus on interval hypergraphs, where all hyperedges are intervals of~$[n]$.
We characterize the interval hypergraphs~$\mathbb{I}$ for which~$P_{\mathbb{I}}$ is a lattice, and if times allows, a distributive lattice, a semidistributive lattice, and a lattice quotient of the weak order. Joint work with Vincent Pilaud.