Facial nested complexes and acyclonestohedra

Abstract

We study nested complexes of building sets on the Las Vergnas face lattices of oriented matroids. Such a nested complex is the face lattice of an oriented matroid, obtained by iterated stellar subdivisions of the positive tope. If the oriented matroid is realizable, the nested complex is isomorphic to the boundary complex of a polytope. We turn this into an explicit and combinatorially meaningful polytopal realization. We prove that the facial nested complex can be embedded as the acyclic subcomplex of the nested complex of a well-chosen boolean building set. In the realizable case, we show that this acyclic subcomplex can be geometrically selected as the section of a nestohedron by the evaluation space of the vector configuration, which we call acyclonestohedron. Our framework generalizes the poset associahedra recently introduced by P. Galashin, from order polytopes to any polytope. Poset associahedra are the graphical acyclonestohedra, and our approach recovers as particular cases the main results of P. Galashin and answers some of his open questions. Besides poset associahedra, our framework unifies various other existing families of nested-like polytopes, such as the simple polytope nestohedra, the hyperoctahedral nestohedra, the design graph associahedra and the permutopermutohedra, to which our palette of results can be directly applied. The core of our construction is the embedding of the facial nested complex inside a boolean nested complex. More generally, we provide conditions that guarantee an embedding between nested complexes over two lattices. For instance, any atomic nested complex has a canonical embedding inside a boolean nested complex. As another application, we embed nested complexes over lattices of faces into nested complexes over lattices of flats, recovering as a particular case, the embedding of the positive Bergman complex into the Bergman complex.