Affine and toric hyperplane arrangements
Richard Ehrenborg, Margaret Readdy, MLE Slone
我们将中心超平面排列的交格和面格之间的Billera-Escherborg-Readdy映射推广到仿射超平面和复曲面超平面排列。对于环面上的安排，我们还推广了Zaslavsky关于区域数的基本结果。
We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.
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The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This article has two objectives: On one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we intro…
We give a formula for computing the characteristic polynomial for certain hyperplane arrangements in terms of the number of bipartite graphs of given rank and cardinality.
Pythagorean Hyperplane Arrangements: Combinatorics of Gain Genericity
针对 Zaslavsky 定义的毕达哥拉斯超平面排列，本文提出新的泛型性概念，并构造可决定所有此类排列组合结构的辅助排列，辅以应用与示例。
This paper studies \emph{Dirichlet arrangements}, a generalization of graphic hyperplane arrangements arising from electrical networks and order polytopes of finite posets. We generalize descriptions of combinatorial features of graphic arrangements to Dirichlet arrangements, including characteristic polynomials and supersolvability. We apply these results to visibility sets of order polytopes and fixed-energy harmonic functions on electrical networks.
The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a hyperplane arrangement can be expressed in terms of its Tutte polynomial. We also show that, even if one is only interested in computing the Tutte polynomial of a graph or a matroid, the theory of hyperplane arrangements provides a powerful finite field method for this compu…
Freeness of hyperplane arrangements associated with gain graphs
本文利用增益图推广了Athanasiadis平移超平面构形与Bailey倾斜超平面构形之间自由性等价的现象。
We give a complete formula for the characteristic polynomial of hyperplane arrangements $\mathcal J_n$ consisting of the hyperplanes $x_i+x_j=1$, $x_k=0$, $x_l=1$, $ 1\leq i, j, k, l\leq n$. The formula is obtained by associating hyperplane arrangements with graphs, and then enumerating central graphs via generating functions for the number of bipartite graphs of given order, size and number of connected components.
The set of chambers of a real hyperplane arrangement may be ordered by separation from some fixed chamber. When this poset is a lattice, Bjorner, Edelman, and Ziegler proved that the chambers are in natural bijection with the biconvex sets of the arrangement. Two families of examples of arrangements with a lattice of chambers are simplicial and supersolvable arrangements. For these arrangements, we prove that the chambers correspond to biclosed sets, a weakening of the biconvex property.
We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which are free, simplicial, factored, or supersolvable. In particular, such an arrangement is free if and only if the graph is a cycle, a path, an almost path, or a path with a triangle attached to it.
Bruhat order, smooth Schubert varieties, and hyperplane arrangements
本文的目的是将旗流形中的舒伯特簇与超平面排列联系起来。对于一个置换，我们构造了一个特定的图超平面排列。我们表明，这种安排的区域的生成函数与相应的舒伯特品种的庞加莱多项式相一致，当且仅当舒伯特品种是光滑的。给出了Poincare多项式的一个显式组合公式。我们的主要技术工具是弦图和完全消去序。
On the division of space by topological hyperplanes
拓扑超平面是R^n的一个子空间（或它的同胚），它在拓扑上等价于一个普通的直超平面。R^n中拓扑超平面的排列是一个有限集H，使得H中的k个拓扑超平面，如果它们的交集非空，则在一个子空间中相遇，该子空间是其中任意k-1个拓扑超平面的交集中的拓扑超平面;但是两个相交的拓扑超平面不需要彼此相交。如果每一对相交的线都相交，那么这种排列就是仿射的。由拓扑超平面的排列形成的区域的数量具有与仿射超平面的排列相同的公式。为了从几何学上解释这一点，我们提出了一个问题，即在任何排列中的拓扑超平面的一部分是否可以重新组合成具有相同区域的仿射拓扑超平面的排列。如果维度是2，这总是可能的，但在更高的维度中就不可能了。我们还问是否所有的仿射拓扑超平面安排对应于定向拟阵;他们不需要，但我们可以刻画那些做，如果维度是2。在更高的维度中，这个问题是开放的。另一个悬而未决的问题是刻画拓扑超平面排列的交半格;第三个问题是证明拓扑超平面排列的区域必然是胞。
We prove that for any finite real hyperplane arrangement the average projection volumes of the maximal cones is given by the coefficients of the characteristic polynomial of the arrangement. This settles the conjecture of Drton and Klivans that this held for all finite real reflection arrangements. The methods used are geometric and combinatorial. As a consequence we determine that the angle sums of a zonotope are given by the characteristic polynomial of the order dual of the intersection lattice of the arrangement.
A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes
Shi排列${\mathcal S}_n$是${\mathbb R}^n$中仿射超平面的排列，形式为$x_i - x_j = 0$或$1$，用于$1 \leq i < j \leq n$。它将${\mathbb R}^n$分为$(n+1)^{n-1}$区，这是Shi首先证明的。我们给出了这个结果的一个简单的双射证明。我们的双射很容易推广到任何子安排的${\mathcal S}_n$包含超平面$x_i - x_j = 0$和扩展的施安排。
We consider arrangements of n hyperplanes of codimension one in a real projective space of dimension d. Let us denote by F the maximal possible number f of connected components of the complement in the projective space of dimension d to the union of n hyperplanes. We prove that for sufficiently large n and for d>3 almost all integers between n and F could be realized as the numbers of regions for some arrangements of n hyperplanes in the projective space. This fact was known before for d=2,3.
We exhibit a one-to-one correspondence between $3$-colored graphs and subarrangements of certain hyperplane arrangements denoted $\mathcal J_n$, $n \in \mathbb N$. We define the notion of centrality of $3$-colored graphs which corresponds to the centrality of hyperplane arrangements. Via the correspondence, the characteristic polynomial $χ_{\mathcal J_n}$ of $\mathcal J_n$ can be expressed in terms of the number of central $3$-colored graphs, and we compute $χ_{\mathcal J_n}$ for $n = 2, 3$.
Computing characteristic polynomials of hyperplane arrangements with symmetries
本研究提出一种利用对称群计算超平面构形特征多项式的新算法，可同时统计腔室数量，基于OSCAR实现了julia版本，并通过多领域示例验证了适用性。
Every subarrangement of Weyl arrangements of type $ B_{\ell} $ is represented by a signed graph. Edelman and Reiner characterized freeness of subarrangements between type $ A_{\ell-1} $ and type $ B_{\ell} $ in terms of graphs. Recently, Suyama and the authors characterized freeness for subarrangements containing Boolean arrangements satisfying a certain condition. This article is a sequel to the previous work. Namely, we give a complete characterization for freeness of arrangements between Boolean arrangements and Weyl arrangements of type $ B_{\ell} $ in terms of graphs.
We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken-circuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements.
Computing the Tutte polynomial of a hyperplane arrangement
定义并研究了超平面排列的Tutte多项式。我们介绍了一种方法来计算它通过解决一个有限域中的枚举问题。对于特定的安排，Tutte多项式的计算，然后减少到某些相关的枚举问题。因此，我们得到了新的公式的生成函数枚举交错树，标号树，半序和Dyck路。
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。
We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the Möbius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six.
In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac showed that a graph is chordal if and only if the graph is obtained by "gluing" complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrang…
We find explicit eigenvectors for the transition matrix of a random walk due to Bidegare, Hanlon and Rockmore. This is accomplished by using Brown and Diaconis' analysis of its stationary distribution, together with some combinatorics of functions on the face lattice of a hyperplane arrangement, due to Gelfand and Varchenko.
Tropical hyperplane arrangements and oriented matroids
研究了一类热带超平面构形的组合性质。我们定义了热带定向拟阵，并证明了它们具有普通定向拟阵的许多性质。我们证明了一个热带定向拟阵决定了两个单形的乘积的一个细分，并猜想这种对应是一个双射。
Characteristic elements of the Tits algebra of a real hyperplane arrangement carry information about the characteristic polynomial. We present this notion and its basic properties, and apply it to derive various results about the characteristic polynomial of an arrangement, from Zaslavsky's formulas to more recent results of Kung and of Klivans and Swartz. We construct several examples of characteristic elements, including one in terms of intrinsic volumes of faces of the arrangement.
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。
Free hyperplane arrangements associated to labeled rooted trees
每个标记的有根树与超平面排列相关联，该超平面排列不受由该树的顶点的深度给出的指数的影响。这些安排的交叉格描述通过偏序集的森林。这些偏序集被用来定义余代数，其对偶代数被证明有一个简单的表示的生成元和关系。
Periodicity of hyperplane arrangements with integral coefficients modulo positive integers
研究了整系数模正整数$q$的中心超平面排列.我们证明了基数的补充超平面是一个拟多项式在两个方面，首先通过理论的初等因子，然后通过理论的Ehrhart拟多项式。这一结果对于确定相应实排列的特征多项式是有用的。利用前一种方法，我们还证明了除有限个$q$外，模$q$的交格都是周期的.
For every irreducible complex representation~$π_λ$ of the symmetric group~$§_n$, we construct, in a canonical way, a so-called intrinsic hyperplane arrangement~$\A_λ$ in the space of~$π_λ$. This arrangement is a direct generalization of the classical braid arrangement (which is the special case of our construction corresponding to the natural representation of~$§_n$), has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties.
Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements
形式为x_j = x_i + c的超平面称为仿射平面.对于R^n中的仿射超平面排列，例如Shi排列，我们研究了函数f（M），它计数[1，M]^n中不位于该排列的任何超平面内的整数点。我们证明了f（M）是正整数M的分段多项式函数，由随着M的增加逐渐出现的项组成.我们的方法是转换的问题之一，计数积分正确着色的根积分增益图。应用于区间着色，其中顶点v_i的可用颜色的区间具有[（h_i）+1，M]的形式.一个相关的问题需要颜色模M;适当的模块着色的数量是一个不同的分段多项式，对于大M成为特征多项式的安排（这意味着Athanasiadis以前获得的多项式）。我们还研究了这个功能的所有积极的模。
Originally in 1954 the Tutte polynomial was a bivariate polynomial associated to a graph in order to enumerate the colorings of this graph and of its dual graph at the same time. However the Tutte polynomial reveals more of the internal structure of a graph, and contains even other specializations from other sciences like the Jones polynomial in Knot theory, the partition function of the Pott model in statistical physics, and the reliability polynomial in network theory. In this article, we study the Tutte polynomial associated to more general objects which are the arrangements of hyperplanes.…
We characterize in simple terms the Pak-Stanley labels $λ(R)$ of the regions $R$ of the $m$-Catalan arrangement. We also propose a simple algorithm that returns $R$ from $λ(R)$. Finally, we characterize in close terms the labels of the relatively bounded regions.
Face Counting for Topological Hyperplane Arrangements
本文针对Forge与Zaslavsky提出的拓扑超平面配置结构，计算了横贯型拓扑超平面配置的f-多项式，进而推导出其腔室数量，拓展了经典超平面配置面计数的相关研究结果。
Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements
单调路径多面体是由Billera和Sturmfels提出的纤维多面体的一个特例。一个简单的例子是置换面体，它是标准单位立方体的单调路径多面体。置换面体是相对于辫状排列的极带形。我们展示了如何zonotopes极锥的某些变形的编织安排可以实现为单调的路径多面体。建设是一个扩展的permutahedron和产生有趣的连接枚举组合的超平面安排和几何单调路径多面体。
Ardila and Develin's paper on tropical oriented hyperplane arrangements and tropical oriented matroids defines tropical oriented matroids and conjectures a bijection between them and triangulations of products of simplices $Δ_{n-1} \times Δ_{d-1}$. Oh and Yoo recently confirmed this conjecture; however, neither group addressed the case of hyperplanes that are not in generic position. These non-generic arrangements do not correspond to tropical oriented matroids, but they encode information about subdivisions of $Δ_{n-1} \times Δ_{d-1}$. This note considers the non-generic case and presents som…
We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the $n$-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defi…
A combinatorial reciprocity theorem for hyperplane arrangements
给定一个非负整数$m$和${\mathbb Q}^d$上线性型的有限集合${\mathcal A}$，由$α\in {\mathcal A}$和整数$k \in [-m, m]$的方程$α(x) = k$定义的${\mathbb Q}^d$中仿射超平面的排列表示为${\mathcal A}^m$。证明了${\mathcal A}^m$的特征多项式的系数在$m$中是拟多项式，并且满足一个简单的组合互反律.
Symmetry of $f$-vectors of toric arrangements in general position and some applications
本文研究环面上一般位置有限超平面排列的组合性质，刻画了其 f-向量的对称性，并将该结果应用于超平面构型的计数问题。
Simplex volumes in hyperplane arrangements
该研究探讨了超平面排列中简单体积的问题，包括单位体积简单体积的最大数目、最小/最大体积简单体积的最大数目以及不同体积简单体积的排列问题。
One-element Extensions of Hyperplane Arrangements
本文利用诱导伴随构形分类超平面构形的单元素扩张，证明多种组合不变量在相关层上恒定且保序，并得到有限域或有理情形下特征多项式的卷积公式。
An element $w$ of the Weyl group is called rationally smooth if the corresponding Schubert variety is rationally smooth. This happens exactly when the lower interval $[id,w]$ in the Bruhat order is palindromic. For each element $w$ of the Weyl group, we construct a certain hyperplane arrangement. After analyzing the palindromic intervals inside the maximal quotients, we use this result to show that the generating function for regions of the arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth.
Flips of diagonals in colored triangle-free triangulations of a convex polygon are interpreted as moves between two adjacent chambers in a certain graphic hyperplane arrangement. Properties of geodesics in the associated flip graph are deduced. In particular, it is shown that: (1) every diagonal is flipped exactly once in a geodesic between distinguished pairs of antipodes; (2) the number of geodesics between these antipodes is equal to twice the number of Young tableaux of a truncated shifted staircase shape.
This paper provides an overview of selected results and open problems in the theory of hyperplane arrangements, with an emphasis on computations and examples. We give an introduction to many of the essential tools used in the area, such as Koszul and Lie algebra methods, homological techniques, and the Bernstein-Gelfand-Gelfand correspondence, all illustrated with concrete calculations. We also explore connections of arrangements to other areas, such as De Concini-Procesi wonderful models, the Feichtner-Yuzvinsky algebra of an atomic lattice, fatpoints and blowups of projective space, and plan…
Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley
如果一个超平面的特征多项式的所有根都有相同的实部，则称该超平面的排列满足"黎曼假设"。这一性质是由Postnikov和Stanley为某些家庭的安排，这是定义为任何不可约的根系，并证明了根系$A_{n-1}$。证明是基于一个明确的公式的特征多项式，这是独立的组合意义。这里我们以前推导这个公式是简化和扩展到类似的公式，但例外的根系。在这些情况下，我们可以推测。
Deformations of Coxeter hyperplane arrangements
我们研究了几个超平面安排，可以被视为变形的Coxeter安排。特别地，我们证明了Linial和Stanley的一个猜想，即排列x_i - x_j = 1，1 \leq i<j \leq n的区域数等于n+1个顶点上交错树的数目.值得注意的是，这些数字在二叉树、偏序集和竞赛方面有几个额外的组合解释。更一般地，我们给出了A_{n-1}型仿射Coxeter排列的某些有限子排列的区域数和Poincar多项式的公式。这些公式使我们能够证明一个关于Poincar多项式零点位置的“黎曼假设“。我们还考虑了A_{n-1}型Coxeter安排的一些一般变形。
Terao's factorization theorem shows that if an arrangement is free, then its characteristic polynomial factors into the product of linear polynomials over the integer ring. This is not a necessary condition, but there are not so many non-free arrangements whose characteristic polynomial factors over the integer ring. On the other hand, the localization of a free arrangement is free, and its restriction is in many cases free, thus its characteristic polynomial factors. In this paper, we consider how their integer, or real roots behave.
Flats and hyperplane arrangements for matroids with coefficients
本文提出了一种基于 tract 的 matroid 理论，通过 flats 和超平面排列等概念，扩展了 matroid 的描述方式，并在热带线性空间中进行了验证。
A toric arrangement is an arrangement of subtori of codimension one in a real or complex torus. The poset of layers is the set of connected components of non-empty intersections of these subtori, partially ordered by reverse inclusion. In this note we present an algorithm that computes this poset in the central case.
Holm introduced $m$-free $\ell$-arrangements which is a generalization of free arrangements, while he asked whether all $\ell$-arrangements are $m$-free for $m$ large enough. Recently Abe and the author verified that this question is in the negative when $\ell\geq 4$. In this paper we verify that $3$-arrangements $\mathscr{A}$ are $m$-free and compute the $m$-exponents for all $m\geq |\mathscr{A}|+2$, where $|\mathscr{A}|$ is the cardinality of $\mathscr{A}$. Hence Holm's question is in the positive when $\ell=3$. Finally we prove that $3$-dimensional Weyl arrangements of types A and B are $m$…
Topological complexity of generic hyperplane arrangements
证明了$r$维线性空间中$n$超平面的一般复本质超平面排列的补的拓扑复杂度为min $\{n+1,2r\}$ .
We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin in their definition of a tropical oriented matroid, induces an action of the hyperplane face monoid of the classical braid arrangement on the arrangement, and hence on a number of interesting related structures. Along the way, we introduce a new characterization of the types (in the sense of Develin and Sturmfels) of points with respect to a tropica…
We introduce the package \textbf{arrangements} for the software CoCoA. This package provides a data structure and the necessary methods for working with hyperplane arrangements. In particular, the package implements methods to enumerate many commonly studied classes of arrangements, perform operations on them, and calculate various invariants associated to them.
Strata of toric hyperplane arrangements, zonotope lattice points, and the Bondal-Thomsen collection
本文证明了有向环面超平面排列的层与带形多胞形格点间的双射，并关联了层维数与最小面维数，进而探讨了与环面簇及Bondal-Thomsen生成元的联系。
Generic section of a hyperplane arrangement and twisted Hurewicz maps
我们考虑一个扭曲的版本的Hurewicz地图上的补充超平面安排。本文的目的是证明扭曲Hurewicz映射在某些一般性条件下的满射性。作为推论，我们还证明了超平面排列的补的一般截面具有非平凡的同伦群。
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $σ$-Gröbner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals.
This is the expanded notes of the lecture by the author in "Arrangements in Pyrenees", June 2012. We are discussing relations of freeness and splitting problems of vector bundles, several techniques proving freeness of hyperplane arrangements, K. Saito's theory of primitive derivations for Coxeter arrangements, their application to combinatorial problems and related conjectures.
We prove the existence of Gysin morphisms for hyperplane sections that may not satisfy the usual hypotheses of the Lefschetz hyperplane theorem. As an application, we show the triviality of the Alexander polynomial of a particular class of non-symmetric line arrangements, thus providing positive evidence for a conjecture of Papadima and Suciu.
Descent algebras, hyperplane arrangements, and shuffling cards
给出了有限Coxeter群上的两个Riffle洗牌的概念：一个是利用Solomon的下降代数，另一个是利用超平面排列的腔室上的随机游动。这些与$A$、$B$、$C$、$H_3$类型一致，并分为两组。这两个概念都有相同的简单特征值。当$W$是晶体学的并且相关参数是良素数时，超平面定义是特别自然的并且满足正性性质。超平面的观点提出了有趣的联系与李理论，并导致了一个概念的riffle洗牌任意实超平面安排和定向拟阵。与切利尼的下降代数的连接。
Hyperplane arrangements form the latest addition to the zoo of combinatorial objects dealt with by polymake. We report on their implementation and on a algorithm to compute the associated cell decomposition. The implemented algorithm performs significantly better than brute force alternatives, as it requires less convex hulls computations.
Let $(X, A)$ be a nonsingular polarized toric 3-fold. We show that if the adjoint bundle of $A$ has no glabal section, then all ample line bundles on $X$ are normally generated. Even if the adjoint bundle is effective, if it is not big, then it is shown the normal generation. In particular, we show that all ample line bundles on a nonsingular toric Fano 3-fold are normally generated.
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