P-associahedra

Abstract

For each poset P, we construct a polytope \({\mathscr {A}}(P)\) called the P-associahedron. Similarly to the case of graph associahedra, the faces of \({\mathscr {A}}(P)\) correspond to certain nested collections of subsets of P. The Stasheff associahedron is a compactification of the configuration space of n points on a line, and we recover \({\mathscr {A}}(P)\) as an analogous compactification of the space of order-preserving maps \(P\rightarrow {{\mathbb {R}}}\). Motivated by the study of totally nonnegative critical varieties in the Grassmannian, we introduce affine poset cyclohedra and realize these polytopes as compactifications of configuration spaces of n points on a circle. For particular choices of (affine) posets, we obtain associahedra, cyclohedra, permutohedra, and type B permutohedra as special cases.