Lattices from graph associahedra (Q2306513)

Given a simple, finite graph \(G = (V, E)\) and \(I \subseteq V\), let \(G|_I\) denote the induced subgraph with vertex set \(I\). A tube of \(G\) is a nonempty subset \(I \subseteq V\) such that \(G|_I\) is connected. Fixing the standard basis vectors \(e_j\) of \(\mathbb{R}^V\) and given \(I \subseteq V\), let \(\Delta_I\) be the convex hull of \(e_j\) for \(j \in I\). The graph associahedron is the Minkowski (pointwise) sum \[ P_G \ := \sum_{ I \text{ tube of } G } \Delta_I. \] This is an example of a generalized permatohedron, i.e., the normal fan of \(P_G\) is a coarsening of that of the permutohedron (which is the graph associahedron of a complete graph). Given a linear functional \(\lambda\), we can construct a poset \(L_G\) on the vertices of \(P_G\) by taking the reflexive and transitive closure of the relation \(x \preceq y\) when \([x,y]\) is an edge of \(P_G\) and \(\lambda(x) \le \lambda(y)\). Because the normal fan of \(P_G\) coarsens that of the permutohedron, there is a canonical surjection \(\Psi_G\) from the poset (actually, lattice) of the weak order on the symmetric group on \(V\) to \(L_G\). The main result of the paper under review is that \(\Psi_G\) is a lattice quotient map (i.e., it preserves meets and joins) if and only if \(G\) is filled, i.e., for each edge \(ik \in E\), there are edges \(ij \in E\) and \(jk \in E\) whenever \(i < j < k\). (This assumes a fixed order on \(V\).)