Face Poset Research Articles

The discrete flow category: structure and computation

AbstractIn this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined from a discrete Morse function on a regular CW complex. We design an algorithm to efficiently compute the Hom posets of the discrete flow category in this case. Furthermore, we show that in the special case where the discrete Morse function is defined on a simplicial complex, then each Hom poset has the structure of a face poset of a regular CW complex. Finally, we prove that the spectral sequence associated to the double nerve of the discrete flow category collapses on page 2.

Poincaré duality for a class of posets

We prove the Poincaré duality theorem for bi-cellular posets X, that is, both X and Xop are cellular, in terms of cap product for finite posets which will be introduced. Moreover, we show that our results include face posets of h-regular homology manifolds.

Face Posets of Tropical Polyhedra and Monomial Ideals

We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how their geometry reflects properties of monomial ideals. Their vertex-facet lattice is homotopy equivalent to a sphere and encodes the Betti numbers of an associated monomial ideal.

Posets for which Verdier duality holds

We discuss two known sheaf-cosheaf duality theorems: Curry’s for the face posets of finite regular CW complexes and Lurie’s for compact Hausdorff spaces, i.e., covariant Verdier duality. We provide a uniform formulation for them and prove their generalizations. Our version of the former works over the sphere spectrum and for more general finite posets, which we characterize in terms of the Gorenstein* condition. Our version of the latter says that the stabilization of a proper separated infty -topos is rigid in the sense of Gaitsgory. As an application, for stratified topological spaces, we clarify the relation between these two duality equivalences.

Lattice of integer flows and the poset of strongly connected orientations for regular matroids

A 2010 result of Amini provides a way to extract information about the structure of the graph from the geometry of the Voronoi polytope of the lattice of integer flows (which determines the graph up to two-isomorphism). Specifically, Amini shows that the face poset of the Voronoi polytope is isomorphic to the poset of strongly connected orientations of subgraphs. This answers a question raised by Caporaso and Viviani, and Amini also proves a dual result for integer cuts. In this paper we generalize Amini's result to regular matroids; in this context the theorem for integer cuts becomes a direct consequence of the theorem for integer flows, by making duality explicit as matroid duality.

On the f-vectors of r-multichain subdivisions

For a poset P and an integer r≥1, let Pr be the collection of all r-multichains in P. Corresponding to each strictly increasing map ı:[r]→[2r], there is an order ⪯ı on Pr. Let Δ(Gı(Pr)) be the clique complex of the graph Gı associated to Pr and ı. In a recent paper [14], it is shown that Δ(Gı(Pr)) is a subdivision of P for a class of strictly increasing maps. In this paper, we show that all these subdivisions have the same f-vector. We give an explicit description of the transformation matrices from the f- and h-vectors of Δ to the f- and h-vectors of these subdivisions when P is a poset of faces of Δ. We study two important subdivisions, namely Cheeger-Müller-Schrader's subdivision and the r-colored barycentric subdivision which fall in our class of r-multichain subdivisions.

Cluster-Permutohedra and Submanifolds of Flag Varieties with Torus Actions

Abstract In this paper, we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardo, Del Río-Francos, López-Dudet, Oliveros, and Schulte in 2010 and toric topology of manifolds of sparse isospectral Hermitian matrices. More precisely, we recall the notion of a cluster-permutohedron, a certain finite poset defined for a simple graph $\Gamma $. This poset is build as a combination of cosets of the symmetric group, and the geometric lattice of the graphical matroid of $\Gamma $. This poset is similar to the graphicahedron of $\Gamma $, in particular, 1-skeleta of both posets are isomorphic to Cayley graphs of the symmetric group. We describe the relation between cluster-permutohedron and graphicahedron using Galois connection and the notion of a core of a finite topology. We further prove that the face poset of the natural torus action on the manifold of isospectral $\Gamma $-shaped Hermitian matrices is isomorphic to the cluster-permutohedron. Using recent results in toric topology, we show that homotopy properties of graphicahedra may serve an obstruction to equivariant formality of isospectral matrix manifolds. We introduce a generalization of a cluster-permutohedron and describe the combinatorial structure of a large family of manifolds with torus actions, including Grassmann manifolds and partial flag manifolds.

The permuto-associahedron revisited

A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset, determine whether there exists a polytope whose face lattice is the poset. In 1990s, Kapranov defined a poset as a hybrid between the face poset of a permutohedron and that of an associahedron, and he asked whether this poset is realizable. Shortly after his question was posed, Reiner and Ziegler provided a realization. Based on our previous work on the nested braid fan, we provide in this paper a different realization of Kapranov’s poset by constructing the vertex set and the normal fan of a permuto-associahedron simultaneously.

Shelling the \(m=1\) amplituhedron

The amplituhedron $\mathcal{A}_{n,k,m}$ was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is a projection inside the Grassmannian $\text{Gr}_{k,k+m}$ of the totally nonnegative part of $\text{Gr}_{k,n}$. Karp and Williams (2019) studied the $m=1$ amplituhedron $\mathcal{A}_{n,k,1}$, giving a regular CW decomposition of it. Its face poset $R_{n,l}$ (with $l := n-k-1$) consists of all projective sign vectors of length $n$ with exactly $l$ sign changes. We show that $R_{n,l}$ is EL-shellable, resolving a problem posed by Karp and Williams. This gives a new proof that $\mathcal{A}_{n,k,1}$ is homeomorphic to a closed ball, which was originally proved by Karp and Williams. We also give explicit formulas for the $f$-vector and $h$-vector of $R_{n,l}$, and show that it is rank-log-concave and strongly Sperner. Finally, we consider a related poset $P_{n,l}$ introduced by Machacek (2019), consisting of all projective sign vectors of length $n$ with at most $l$ sign changes. We show that it is rank-log-concave, and conjecture that it is Sperner.

Shellable tilings on relative simplicial complexes and their h-vectors

Abstract An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision.

Colorful graph associahedra

Given a graph G, the graph associahedron is a simple convex polytope whose face poset is based on the connected subgraphs of G. With the additional assignment of a color palette, we define the colorful graph associahedron, show it to be a collection of simple abstract polytopes, and explore its properties.

Counting weighted maximal chains in the circular Bruhat order

The totally nonnegative Grassmannian Gr(k,n)≥0 is the subset of the real Grassmannian Gr(k,n) consisting of points with all nonnegative Plücker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of Postnikov's positroid cell decomposition of Gr(k,n)≥0[10]. We provide a closed formula for the sum of its weighted chains in the spirit of Stembridge [11].

Combinatorial and topological aspects of path posets, and multipath cohomology

Multipath cohomology is a cohomology theory for directed graphs, which is defined using the path poset. The aim of this paper is to investigate combinatorial properties of path posets and to provide computational tools for multipath cohomology. In particular, we develop acyclicity criteria and provide computations of multipath cohomology groups of oriented linear graphs. We further interpret the path poset as the face poset of a simplicial complex, and we investigate realisability problems.

Polyhedral products over finite posets

Polyhedral products were defined by Bahri, Bendersky, Cohen, and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper, we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets. We show that under certain restrictions on the poset P, which include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over P of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction—the polyhedral tensor product. We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product. We then restrict attention to polyhedral posets, a family of posets that includes face posets of simplicial complexes, and simplicial posets, as well as many others. We define the Stanley–Reisner ring of a polyhedral poset and show that, as in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset P, we construct a simplicial poset s(P), and show that if P is a polyhedral poset, then polyhedral products over P coincide up to homotopy with the corresponding polyhedral products over s(P).

On the topology of bi-cyclopermutohedra

Motivated by the work of Panina and her coauthors on cyclopermutohedron we study a poset whose elements correspond to equivalence classes of partitions of the set $$\{1,\dots , n+1\}$$ up to cyclic permutations and orientation reversion. This poset is the face poset of a regular CW complex which we call bi-cyclopermutohedron and denote it by $$\mathrm {QP}_{n+1}$$ . The complex $$\mathrm {QP}_{n+1}$$ contains subcomplexes homeomorphic to moduli space of certain planar polygons with $$n+1$$ sides up to isometries. In this article we find an optimal discrete Morse function on $$\mathrm {QP}_{n+1}$$ and use it to compute its homology with $${\mathbb {Z}}$$ as well as $${\mathbb {Z}}_2$$ coefficients.

The edge-product space of phylogenetic trees is not shellable

The edge-product space of phylogenetic trees is a regular CW complex whose maximal closed cells correspond to trivalent trees with leaves labeled by a finite set X. The face poset of this cell decomposition is isomorphic to the Tuffley poset, a poset of labeled forests, with a unique minimum adjoined. We show that the edge-product space of phylogenetic trees is gallery-connected. We then use combinatorial properties of the Tuffley poset and a related graph known as NNI-tree space to show that, although open intervals of the Tuffley poset were proven to be shellable by Gill, Linusson, Moulton, and Steel, the edge-product space is not shellable.

Resolving Stanley's conjecture on $k$-fold acyclic complexes

In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank \(1\) boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.Mathematics Subject Classifications: 05E45, 55U10

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

Schröder combinatorics and [formula omitted]-associahedra

We study ν-Schröder paths, which are Schröder paths which stay weakly above a given lattice path ν. Some classical bijective and enumerative results are extended to the ν-setting, including the relationship between small and large Schröder paths. We introduce two posets of ν-Schröder objects, namely ν-Schröder paths and trees, and show that they are isomorphic to the face poset of the ν-associahedron Aν introduced by Ceballos, Padrol and Sarmiento. A consequence of our results is that the i-dimensional faces of Aν are indexed by ν-Schröder paths with i diagonal steps, and we obtain a closed-form expression for these Schröder numbers in the special case when ν is a ‘rational’ lattice path. Using our new description of the face poset of Aν, we apply discrete Morse theory to show that Aν is contractible. This yields one of two proofs presented for the fact that the Euler characteristic of Aν is one. A second proof of this is obtained via a formula for the ν-Narayana polynomial in terms of ν-Schröder numbers.

Initial degenerations of Grassmannians

We construct closed immersions from initial degenerations of $${{\,\mathrm{Gr}\,}}_{0}(d,n)$$ —the open cell in the Grassmannian $${{\,\mathrm{Gr}\,}}(d,n)$$ given by the nonvanishing of all Plucker coordinates—to limits of thin Schubert cells associated to diagrams induced by the face poset of the corresponding tropical linear space. These are isomorphisms when (d, n) equals (2, n), (3, 6) and (3, 7). As an application we prove $${{\,\mathrm{Gr}\,}}_0(3,7)$$ is schon, and the Chow quotient of $${{\,\mathrm{Gr}\,}}(3,7)$$ by the maximal torus in $$ {\text {PGL}}(7)$$ is the log canonical compactification of the moduli space of 7 points in $${\mathbb {P}}^2$$ in linear general position, making progress on a conjecture of Hacking, Keel, and Tevelev.