Facets of the weak order polytope derived from the induced partition projection (Q2784506)

The weak order polytope \(P^n_{WO}\) in \(\mathbb{R}^{n(n-1)}\) has a vertex \(x^W\) for each weak order \(W\) on \({\mathbf n}:= \{1,\dots, n\}\); similarly defined is the partition polytope \(P^n_{PA}\) in \(\mathbb{R}^{n(n- 1)/2}\); with a vertex \(y^E\) for each equivalence relation \(E\) on \({\mathbf n}\). The mapping \(\pi: W\mapsto E:= W\cap W^{-1}\) induces a corresponding mapping \(\pi^{-1}\), taking faces of \(P^n_{PA}\) to faces of \(P^n_{WO}\); however, it is shown that not all facets of the former lift to facets of the latter. The authors also use this mapping, and known properties of \(P^n_{PA}\), to describe further facets of \(P^n_{WO}\).