Hyperplane arrangements and K-theory
Nicholas J. Proudfoot
本文研究实超平面构形复化的补的Z/2-等变K-理论。计算了有理K环和KO环，给出了由线丛生成的整数KO环的子环的两种不同的组合刻划。
We study the Z/2-equivariant K-theory of the complement of the complexification of a real hyperplane arrangement. We compute the rational K and KO rings, and give two different combinatorial descriptions of the subring of the integral KO ring generated by line bundles.
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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…
Affine and toric hyperplane arrangements
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 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$. The formula is obtained by associating hyperplane arrangements with graphs, and then enumerating central graphs via generating functions for the number of bipartite graphs of given order, size and number of connected components.
The set of chambers of a real hyperplane arrangement may be ordered by separation from some fixed chamber. When this poset is a lattice, Bjorner, Edelman, and Ziegler proved that the chambers are in natural bijection with the biconvex sets of the arrangement. Two families of examples of arrangements with a lattice of chambers are simplicial and supersolvable arrangements. For these arrangements, we prove that the chambers correspond to biclosed sets, a weakening of the biconvex property.
We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which are free, simplicial, factored, or supersolvable. In particular, such an arrangement is free if and only if the graph is a cycle, a path, an almost path, or a path with a triangle attached to it.
Bruhat order, smooth Schubert varieties, and hyperplane arrangements
本文的目的是将旗流形中的舒伯特簇与超平面排列联系起来。对于一个置换，我们构造了一个特定的图超平面排列。我们表明，这种安排的区域的生成函数与相应的舒伯特品种的庞加莱多项式相一致，当且仅当舒伯特品种是光滑的。给出了Poincare多项式的一个显式组合公式。我们的主要技术工具是弦图和完全消去序。
On the division of space by topological hyperplanes
拓扑超平面是R^n的一个子空间（或它的同胚），它在拓扑上等价于一个普通的直超平面。R^n中拓扑超平面的排列是一个有限集H，使得H中的k个拓扑超平面，如果它们的交集非空，则在一个子空间中相遇，该子空间是其中任意k-1个拓扑超平面的交集中的拓扑超平面;但是两个相交的拓扑超平面不需要彼此相交。如果每一对相交的线都相交，那么这种排列就是仿射的。由拓扑超平面的排列形成的区域的数量具有与仿射超平面的排列相同的公式。为了从几何学上解释这一点，我们提出了一个问题，即在任何排列中的拓扑超平面的一部分是否可以重新组合成具有相同区域的仿射拓扑超平面的排列。如果维度是2，这总是可能的，但在更高的维度中就不可能了。我们还问是否所有的仿射拓扑超平面安排对应于定向拟阵;他们不需要，但我们可以刻画那些做，如果维度是2。在更高的维度中，这个问题是开放的。另一个悬而未决的问题是刻画拓扑超平面排列的交半格;第三个问题是证明拓扑超平面排列的区域必然是胞。
Topological complexity of generic hyperplane arrangements
证明了$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 $\{n+1,2r\}$ .
Projection volumes of hyperplane arrangements
证明了对任意有限实超平面排列，极大锥的平均投影体积由该排列的特征多项式的系数给出。这解决了德顿和克利万斯的猜想，即这适用于所有有限的真实反射安排。使用的方法是几何和组合。作为结果，我们确定的角度和zonotope是由特征多项式的顺序对偶的交叉晶格的安排。
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。
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…
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。
On intersection lattices of hyperplane arrangements generated by generic points
我们考虑由泛点生成的超平面排列，并研究它们的交格。这些安排被认为是等同于判别安排。通过将偏序集理想分解为对应于较小维数的较小格的直积，我们给出了交格的基本结构。基于这种分解，我们计算的默比乌斯函数的晶格和特征多项式的安排，直到6维。
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…
Eigenvectors for a random walk on a hyperplane arrangement
我们发现明确的特征向量的转移阵的随机游动由于Bidegare，Hanlon和Rockmore。这是通过使用布朗和Diaconis'分析其平稳分布，连同一些组合的功能面格的超平面安排，由于Gelfand和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.
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.
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.…
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 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。
In this article we describe two new characterizations of freeness for hyperplane arrangements via the study of the generic initial ideal and of the sectional matrix of the Jacobian ideal of arrangements.
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。
In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over $\mathbb{Q}$.
Non-Generic Tropical Hyperplane Arrangements and the Secondary polytope of $Δ_{n-1} \times Δ_{d-1}$
Ardila和Reynin关于热带定向超平面排列和热带定向拟阵的论文定义了热带定向拟阵，并在它们之间建立了一个双射和单形乘积的三角剖分$Δ_{n-1} \times Δ_{d-1}$。Oh和Yoo最近证实了这一猜想;然而，这两个小组都没有解决不在通用位置的超平面的情况。这些非通用的安排不对应于热带定向拟阵，但它们编码的信息细分的$Δ_{n-1} \times Δ_{d-1}$。本说明考虑了非通用的情况下，并在该领域提出了一些初步的结果。
Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes
本文研究了某些组合定义复形的正合性，它们推广了几何格上的Orlik-Solomon代数。主要结果涉及复杂的反射安排及其限制。特别地，我们考虑了对应关系复形，并给出了这些超平面安排的$n$ -形式的简单证明。作为应用，我们能够绑定的Castelnouvo-Mumford正则性的某些模块在多项式环相关的Coxeter安排（实反射安排）和他们的限制。所讨论的模块被定义使用的关系复杂的考克斯特安排和纤维多面体的双重考克斯特zonotope。它们推广了原排列上的分段多项式函数代数。
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.
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.
Freeness of hyperplane arrangements and related topics
这是作者在2012年6月的“比利牛斯山脉的安排”中的演讲的扩展笔记。讨论了向量丛的自由性与分裂问题的关系，证明超平面排列自由性的几种技巧，K。Saito的Coxeter排列的原始导子理论，它们在组合问题中的应用及相关图解。
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.
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.
On homological properties of some Cynk-Szemberg octic hyperplane arrangements
本文研究了Cynk-Szemberg八次超平面排列的同调性质，定义了其类型并提出了本质超平面排列的非自由准则。
This paper studies Markov chains on the chambers of real hyperplane arrangements, a model that generalizes famous examples, such as the Tsetlin library and riffle shuffles. We discuss cutoff for the Tsetlin library for general weights, and we give an exact formula for the separation distance for the hyperplane arrangement walk. We introduce lower bounds, which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions. Using similar techniques, we also prove a uniform lower bound for the mixing time of Glauber dynamics on a monotone system.
Differential Equations for Moving Hyperplane Arrangements
本文研究超平面乘积的Mellin积分（组合关联子）在移动超平面时的行为，并旨在仅基于超平面排列构造全纯湮灭$D$-理想以编码这些函数。
Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux
将凸多边形的着色无三角形三角剖分中对角线的翻转解释为在某一图形超平面排列中两个相邻腔室之间的移动。推导了相应翻转图中测地线的性质。特别是，它表明：（1）每一个对角线是翻转一次，在一个测地线之间的显着对极;（2）这些对极之间的测地线的数量等于两倍的数量的杨氏tableaux截断移位楼梯形状。
The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first formula and asked to give a combinatorial interpretation. In this article, we give a combinatorial formula for the Falk invariant of hyperplane arrangements attached to certain gain graphs.
Hyperplane Arrangements: Computations and Conjectures
本文提供了一个概述的超平面安排理论中的选定结果和开放的问题，重点是计算和例子。我们给出了一个介绍，在该领域中使用的许多重要工具，如Koszul和李代数方法，同调技术，和伯恩斯坦Gelfand-Gelfand对应，都说明了具体的计算。我们还探讨了其他领域的安排，如德Concini-Procesi美妙的模型，Feichtner-Yuzvinsky代数的原子晶格，胖点和爆破的射影空间，和平面曲线奇点的连接。
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。
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