Cubic Partial Cubes from Simplicial Arrangements

Crystallographic arrangements: Weyl groupoids and simplicial arrangements

We introduce the simple notion of a "crystallographic arrangement" and prove a one-to-one correspondence between these arrangements and the connected simply connected Cartan schemes for which the real roots are a finite root system (up to equivalence on both sides). We thus obtain a more accessible definition for this very large subclass of the class of simplicial arrangements for which a complete classification is known.

On simplicial arrangements in $\mathbb{P}^3(\mathbb{R})$ with splitting polynomial

In this paper, we study simplicial hyperplane arrangements in real projective $3$-space. We give a necessary condition for the characteristic polynomial to have only real roots, valid also for non-simplicial arrangements. As application, we obtain combinatorial inequalities which are satisfied for arrangements with splitting polynomial. This allows us to prove that there are only finitely many different isomorphism classes of simply laced simplicial arrangements whose characteristic polynomials split over $\mathbb{R}$. We also provide an updated version of a catalogue published by Grünbaum and…

具有特殊顶点的单纯排列

Simplicial arrangements with special vertex

研究用任意有限超平面排列代替Weyl排列形成的具有特殊顶点的排列，核心方法未提及，主要贡献是证明某些此类单纯排列中直线平行类数量至多为12。

Combinatorial simpliciality of arrangements of hyperplanes

We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present some series of arrangements related to the known arrangements in characteristic zero. We further enumerate simplicial arrangements with given symmetry groups. Finally, we determine all finite complex reflection groups affording combinatorially simplicial arrangements. It turns out that combinatorial simpliciality coincides with inductive freeness for finite c…

单纯超平面配置中超越秩的序维数

Order dimension beyond rank for simplicial hyperplane arrangements

本研究证明单纯超平面配置的区域偏序集序维数可超越其秩，解答了Reading与Segovia的问题，并以Coxeter配置H₄、E₆为例给出具体下界。

q-图安排的图形化扩展到单纯复形

A generalization of $q$-deformation of graphic arrangements to simplicial complexes

本文提出两种新的超平面安排，扩展q-变形到单纯复形，并研究了图形单项式安排的特征多项式和自由性。

单纯集与超可解超平面排列中的哈密顿环

Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements

研究超平面排列的tope图中哈密顿环的存在性，特别是单纯集、反射和超可解排列，并证明所有3维单纯集排列及有限反射排列均存在哈密顿环。

Combinatorics of free and simplicial line arrangements

We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive inequalities involving the $t$-vectors of the arrangements in consideration. As application, we obtain some finiteness and classification results. Moreover, we are able to prove the Dirac Motzkin Conjecture for real pseudoline arrangements whose charateristic polynomials split over $\mathbb{R}$.

Skeleton Simplicial Evaluation Codes

For a subspace arrangement over a finite field we study the evaluation code defined on the arrangement's set of points. The length of this code is given by the subspace arrangement's characteristic polynomial. For coordinate subspace arrangements, the dimension is bounded below by the face vector of the corresponding simplicial complex. The minimum distance is determined for coordinate subspace arrangements where the simplicial complex is a skeleton.

A greedy algorithm to compute arrangements of lines in the projective plane

We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.

具有少量二重点的单纯排列

Simplicial arrangements with few double points

本文通过几何论证证明，在单纯排列中相关三次曲线必可约，从而在二重点线性界限下证实了Grünbaum的渐近分类猜想。

Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements

Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement can be realized as monotone path polytopes. The construction is an extension of that of the permutahedron and yields interesting connections between enumerative combinatorics of hyperplane arrangements a…

重新审视实射影三维空间中的单形排列

Simplicial arrangements in real projective three-space revisited

本文研究实射影三维空间中的不可约单形排列，构建单形性判据，明确秩4晶体学Coxeter排列的特殊顶点性质，对比不同缺陷并补充相关实例。

On Limits of Dense Packing of Equal Spheres in a Cube

We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of $\lceil p^{3}/2\rceil$ spheres. For this family of packings, the previous best-known arrangements were usually derived from a ccp by omission of a certain number of spheres without changing the initial structure. In this paper, we show that better arrangements exist for all $n\leq\lceil p^{3}/2\rceil-2$. We introduce an optimization method to reveal improvements of these packings, and present many new improvements for $n\leq4629$.

单纯伪超平面排列给出弱Garside群

Simplicial pseudohyperplane arrangements give weak Garside groups

本文通过结合Bessis的Garside范畴与Salvetti的度量半球复形，证明了伪超平面排列复化补集及定向拟阵的Salvetti复形基本群均为弱Garside群，提供了该群的新实例。

图排列的向量场与单纯偏序集的面环

Vector fields of graphic arrangements and face rings of simplicial posets

该研究针对连通简单图对应的图排列，将其对数向量场经微小修改后与单纯偏序集的面环建立同构，进而推导其代数不变量公式并给出显式向量空间基。

Corners and simpliciality in oriented matroids and partial cubes

Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via Metric Graph Theory. Our first main result is that every element of an OM from a class introduced by Mandel is incident to a simplicial tope, i.e, such OMs contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas' simplex conjecture. Further, we show that the mutation graph of un…

Canonical Sphere Bases for Simplicial and Cubical Complexes

Sphere-bases are constructed for the $\mathbb{Z}_2$ vector space formed by the $k$-dimensional subcomplexes, of $n$-simplex (or $n$-cube), for which every $(k{-}1)$-face is contained in a positive even number of $k$-cells; addition is symmetric difference of the corresponding sets of $k$-cells. The bases consist of the boundaries of an algorithmically-specified family of $k{+}1$-simplexes or $k{+}1$-cubes. Geometric properties of these bases are investigated.

Stanley-Reisner rings for symmetric simplicial complexes, G-semimatroids and Abelian arrangements

We extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length (possibly infinite) simplicial posets with a group action. The action on the complex induces an action on the face ring, and we prove that the ring of invariants is isomorphic to the face ring of the quotient simplicial poset under a mild condition on the group action. We also identify a class of actions on simplicial complexes that preserve the homotopical Cohen-Macaulay property under quotients. When the acted-upon poset is the independence complex of a semimatroid, the $h$-polynomia…

The simplicial boundary of a CAT(0) cube complex

For a CAT(0) cube complex $\mathbf X$, we define a simplicial flag complex $\partial_Δ\mathbf X$, called the \emph{simplicial boundary}, which is a natural setting for studying non-hyperbolic behavior of $\mathbf X$. We compare $\partial_Δ\mathbf X$ to the Roller, visual, and Tits boundaries of $\mathbf X$ and give conditions under which the natural CAT(1) metric on $\partial_Δ\mathbf X$ makes it (quasi)isometric to the Tits boundary. $\partial_Δ\mathbf X$ allows us to interpolate between studying geodesic rays in $\mathbf X$ and the geometry of its \emph{contact graph} $Γ\mathbf X$, which is …

关于反射排列限制的伽马向量与周多项式

On gamma-vectors and Chow polynomials of restrictions of reflection arrangements

本文研究反射排列及其限制的多项式不变量，证明反射排列限制的伽马正性，给出B型周多项式的显式组合公式，并证明D型中间排列的h多项式和周多项式具有B型与D型不变量间的线性插值算术性质。

Decomposition of number arrangements in the cube

A subset $M \subset \textbf{R}^3$ is called a \emph{basic subset}, if for any funciton $f \colon M \to \textbf{R}$ there exist such functions $f_1; f_2; f_3 \colon \textbf{R} \to \textbf{R}$ that $f(x_1, x_2, x_3) = f_1(x_1) + f_2(x_2) + f_3(x_3)$ for each point $(x_1, x_2, x_3)\in M$. In this article we prove a criterion for a basic subset for some specific subsets in terms of some graph properties. We also introduce several constructions for mimimal non-basic subsets. The article is written in Russian.

A comparison of Vassiliev and Ziegler-Zivaljevic models for homotopy types of subspace arrangements

In this paper we represent the Vassiliev model for the homotopy type of the one-point compactification of subspace arrangements as a homotopy colimit of an appropriate diagram over the nerve complex of the intersection semilattice of the arrangement. Furthermore, using a generalization of simplicial collapses to diagrams of topological spaces over simplicial complexes, we construct an explicit deformation retraction from the Vassiliev model to the Ziegler-Zivaljevic model.

Combinatorial remarks on a classical theorem of Deligne

We examine Deligne's classical proof of the asphericity of simplicial arrangements from the viewpoint of the combinatorics of the poset of regions of the arrangement. This turns out to be very natural. In particular, we show that an arrangement is simplicial only if it satisfies Deligne's property on positive paths, thus answering a question posed by Luis Paris.

Link complexes of subspace arrangements

Given a simplicial hyperplane arrangement H and a subspace arrangement A embedded in H, we define a simplicial complex Delta_{A,H} as the subdivision of the link of A induced by H. In particular, this generalizes Steingrimsson's coloring complex of a graph. We do the following: (1) When A is a hyperplane arrangement, Delta_{A,H} is shown to be shellable. As a special case, we answer affirmatively a question of Steingrimsson on coloring complexes. (2) For H being a Coxeter arrangement of type A or B we obtain a close connection between the Hilbert series of the Stanley-Reisner ring of Delta_{A,…

Torus actions, equivariant moment-angle complexes, and coordinate subspace arrangements

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in m-dimensional complex space is isomorphic to the cohomology algebra of Stanley-Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) Then we calculate the latter cohomology algebra by means of the standard Koszul resolution of polynomial ring. To prove these facts we construct an equivariant with respect to the torus action homotopy equivalence between the complement of a coordinate subspace arrangement and th…

Shellable complexes and topology of diagonal arrangements

We prove that if a simplicial complex is shellable, then the intersection lattice for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on the data of shelling. Also, we give some examples of diagonal arrangements where the complement is K(π,1), coming from rank 3 matroids.

任意单纯多胞体的面格是哈密顿的

The face lattice of any simplicial polytope is Hamiltonian

本文证明了任意单纯多胞体的面格具有哈密顿性，通过双向线壳化分解面格为立方体，并应用Gregor定理构造哈密顿圈，证实了相关猜想的主要部分。

Reconstructing $d$-manifold subcomplexes of cubes from their $(\lfloor d/2 \rfloor + 1)$-skeletons

In 1984, Dancis proved that any $d$-dimensional simplicial manifold is determined by its $(\lfloor d/2 \rfloor + 1)$-skeleton. This paper adapts his proof to the setting of cubical complexes that can be embedded into a cube of arbitrary dimension. Under some additional conditions (for example, if the cubical manifold is a sphere), the result can be tightened to the $\lceil d/2 \rceil$-skeleton when $d \geq 3$.

The Salvetti Complex and the Little Cubes

We study how the combinatorial structure of the Salvetti complexes of the braid arrangements are related to homotopy theoretic properties of iterated loop spaces. We prove the skeletal filtrations on the Salvetti complexes of the braid arrangements give rise to the cobar-type Eilenberg-Moore spectral sequence converging to the homology of $Ω^2Σ^2 X$. We also construct a new spectral sequence that computes the homology of $Ω^{\ell}Σ^{\ell} X$ for $\ell>2$ by using a higher order analogue of the Salvetti complex. The $E^1$-term of the spectral sequence is described in terms of the homology of $X…

A triangulation of $\CC P^3$ as symmetric cube of $S^2$

The symmetric group $S_3$ acts on $S^2 \times S^2 \times S^2$ by coordinate permutation, and the quotient space $(S^2 \times S^2 \times S^2)/S_3$ is homeomorphic to the complex projective space $\CC P^3$. In this paper, we construct an 124-vertex simplicial subdivision $(S^2 \times S^2 \times S^2)_{124}$ of the 64-vertex standard cellulation $S^2_4 \times S^2_4 \times S^2_4$ of $S^2 \times S^2 \times S^2$, such that the $S_3$-action on this cellulation naturally extends to an action on $(S^2 \times S^2 \times S^2)_{124}$. Further, the $S_3$-action on $(S^2 \times S^2 \times S^2)_{124}$ is "goo…

Covering Partial Cubes with Zones

A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite po…

Two-dimensional partial cubes

We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variet…

面向任意高维扩展器的KKL定理

Toward a KKL Theorem for any HDX

本文提出一种局部到全局方法，证明任何链接满足KKL定理的单纯复形整体也满足KKL型定理，并给出组合高维扩展器上非扩展函数的首次刻画。