Miguel A. Marco-Buzunariz
We give a combinatorial characterization of isotropic subspaces in the Orlik- Solomon algebra of a hyperplane arrangement in terms of decorations of its intersection lattice. We then use this characterization to prove a result that relates these isotropic subspaces with linear systems supported on the arrangement, for arrangements with isolated non-normal crossings of a particular form.
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Given a matroid $M$ one can define its Orlik-Solomon algebra $OS(M)$ and the Bergman fan $Σ_0(M)$. On the other hand to any rational polyhedral fan $Σ$ one can associate its tropical homology and cohomology groups $\F_\bullet(Σ)$, $\F^\bullet (Σ)$. We will show that the projective Orlik-Solomon algebra $OS_0(M)$ is canonically isomorphic to $\F^\bullet (Σ_0(M))$.
We relate the cohomology of the Orlik-Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik-Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in an arbitrary complex rank one local system. We also establish the relationship between the cohomology support loci of the complement of a discriminantal arrangement and the resonant varieties of its Orli…
We give a vanishing theorem for the monodromy eigenspaces of the Milnor fibers of complex line arrangements. By applying the modular bound of the local system cohomology groups given by Papadima-Suciu, the result is deduced from the vanishing of the cohomology of certain Aomoto complex over finite fields. In order to prove this, we introduce degeneration homomorphisms of Orlik-Solomon algebras.
In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids.
We introduce a basis of the Orlik-Solomon algebra labeled by chambers, so called chamber basis. We consider structure constants of the Orlik-Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen's minimal complex from the Aomoto complex.
A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C}, any circuit contained in the union of the pair is also in C. The pair (M,C) can be seen as a matroidal generalization of a biased graph. We introduce and study an Orlik-Solomon type algebra determined by (M,C). If C is the set of all circuits of M this algebra is the Orlik-Solomon algebra of M.
We prove the torsion freeness of the decomposable Orlik--Solomon algebra of a simple matroid on ground set $[n]$. In the class of hypersolvable \& non-supersolvable complex hyperplane arrangements, the torsion freeness, in a certain degree, of this combinatorially defined object, associated to the intersection lattice of the arrangement, impacts on the first non-vanishing higher homotopy group of the complement of the arrangement.
Orlik--Solomon sheaf homology of geometric lattices
该研究将奥利克 - 所罗门层与有限几何格关联，计算其层同调，证明同调集中在最高次且有特定分解，其对均匀拟阵的应用能给出对称群的有趣表示。
We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the co…
The cohomology on the complement of hyperplanes with the coefficients in the rank one local system associated to a generic weight vanishes except in the highest dimension. In this paper, we construct matroids or arrangements and its weights with non-vanishing cohomology of Orlik-Solomon algebras, using decomposable relations arising from Latin hypercubes.
The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal I(M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called X-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To …
A deformation of the Orlik-Solomon algebra of a matroid M is defined as a quotient of the free associative algebra over a commutative ring R with 1. It is shown that the given generators form a Groebner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as R-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Groebner basis and hence the algebra is Koszul
The Orlik-Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first we study homological properties of E-modules as e.g. complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik-Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik-Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the idea…
Matroid complexes and Orlik-Solomon algebras
本文为超可解拟阵的Orlik-Solomon代数构造组合拟自由微分分次模型，推广Kontsevich的容许图cdga，并据此证明其Koszul性。
To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, Andras Szenes has introduced a particular type of bases, the so called "diagonal basis". We prove that this definition extends naturally to a large class of algebras, the so called chi-algebras. Our definitions make also use of an "iterative residue formula" based on the matroidal operation of contraction. This formula can be seen as the combinatorial analogue of an iterative residue formula introduced by Szenes. As an application we deduce nice formulas to express a pure element in a diagonal…
Given a real arrangement $A$, the complement $M(A)$ of the complexification of $A$ admits an action of $\mathbb{Z}_2$ by complex conjugation. We define the equivariant Orlik-Solomon algebra of $A$ to be the $\mathbb{Z}_2$-equivariant cohomology ring of $M(A)$ with coefficients in $\mathbb{Z}_2$. We give a combinatorial presentation of this ring, and interpret it as a deformation of the ordinary Orlik-Solomon algebra into the Varchenko-Gel'fand ring of locally constant $\mattbb{Z}_2$-valued functions on the complement $C(A)$ of $A$ in $\mathbb{R}^n$. We also show that the $\mathbb{Z}_2$-equivar…
A Leray model for the Orlik-Solomon algebra
该研究构造了拟阵上Leray模型的组合推广形式，通过组合爆破数据给出双分次交换微分分次代数的表示，利用Gröbner基构造单项式基，证明其与拟阵的经典Orlik-Solomon代数拟同构。
The paper provides a combinatorial method to decide when the space of local systems with non vanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik-Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg-Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned p…
The Orlik--Solomon algebra of a locally geometric poset and cohomology of complex abelian arrangements
该研究构造了局部几何偏序集的Orlik–Solomon代数，利用其与复阿贝尔李群子群排列的关联，结合拓扑信息计算排列补集的有理上同调，尤其给出紧李群情形下的显式微分分次代数。
Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements
该研究否定了所有Koszul Orlik–Solomon代数都来自超可解构形的猜想，通过构造秩≥3、在ℚ上可实现的不可约非超可解构形，得到了对应的Koszul Orlik–Solomon代数。
The Orlik-Solomon algebra ${\cal A}(G)$ of a matroid $G$ is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing $G$. On the other hand, some features of the matroid $G$ are reflected in the algebraic structure of ${\cal A}(G)$. In this mostly expository article, we describe recent developments in the construction of algebraic invariants of ${\cal A}(G)$. We develop a categorical framework for the statement and proof of recently discovere…
Let A be a (central) arrangement of hyperplanes in a finite dimension complex vector space V. Let M(A) be the dependence matroid determined by A. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The algebra OS(M) is isomorphic to the cohomology algebra of the complement in V of the union of the hyperplanes of A. The Tutte polynomial T(x,y) of M is a powerful invariant of the matroid M. When M(A) is a rank three matroid and A is the complexification of a real arrangement, we prove that OS(M) determines T(x,y).…
On invariant subrings of Orlik--Solomon and Varchenko--Gel'fand algebras in type A
本文研究了类型A的反射排列作用下Orlik-Solomon代数和Varchenko-Gel'fand环的不变子环，提供生成元和关系的简洁表示，并探讨了混合配置空间的上同调结构。
Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three
本文证明了Falk关于秩为3的简单拟阵的Orlik-Solomon代数确定其Tutte多项式的猜想，且该结论对所有秩不超过3的简单拟阵均成立。
We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont's rational formality result to formality over $\mathbb Z$. The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the ar…
We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Orlik-Solomon algebra with the connection operator. Using this result, we establish the relationship between the cohomology support loci of the complement and the resonance varieties of the Orlik-Solomon algebra for any arrangement, and show that the latter are unions of subspace arrangements in general, resolving a conjecture of Falk. We also obtain lower bounds f…
A bi-arrangement of hyperplanes in a complex affine space is the data of two sets of hyperplanes along with a coloring information on the strata. To such a bi-arrangement, one naturally associates a relative cohomology group, that we call its motive. The motivation for studying such relative cohomology groups comes from the notion of motivic period. More generally, we suggest the systematic study of the motive of a bi-arrangement of hypersurfaces in a complex manifold. We provide combinatorial and cohomological tools to compute the structure of these motives. Our main object is the Orlik-Solom…
Generalized Weight Polynomials of Codes through Flats and Orlik-Solomon Algebras of Matroids
本文提出多种从平展格、Orlik-Solomon代数及循环平展格计算拟阵广义重量多项式的方法，并给出线性码广义重量谱的应用及示例。
We study the notion of a nice partition or factorization of a hyperplane arrangement due to Terao from the early 1990s. The principal aim of this note is an analogue of Terao's celebrated addition-deletion theorem for free arrangements for the class of nice arrangements. This is a natural setting for the stronger property of an inductive factorization of a hyperplane arrangement by Jambu and Paris. In addition, we show that supersolvable arrangements are inductively factored and that inductively factored arrangements are inductively free. Combined with our addition-deletion theorem this leads …
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group $W$ afforded by the homogeneous components of its Orlik-Solomon algebra as sums of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture also relates the Orlik-Solomon characters above to the terms of a decomposition of the regular character of $W$ related to the descent algebra of $W$. A consequence of our conjecture is that both the regular character of $W$ and the character of the Orlik-Solomon algebra have parallel, graded de…
The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement A in C^n; it is the quotient of an exterior algebra E(V) on |A| generators. Orlik and Terao introduced a commutative analog S(V)/I of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert series depends only on the intersection lattice L(A). Motivated by topological considerations, Falk and Randell introduced the property of 2-formality; we study the relation between 2-formality and the Orlik-Terao algebra. Our main result is a necessary and sufficient condition for 2-formal…
Matroids, Feynman categories, and Koszul duality
该研究引入并研究特定费曼范畴，将拟阵的Chow环、Orlik–Solomon代数等组合不变量组装为该范畴上的操作子结构，建立二者的Koszul型对偶以推广Getzler的结果，并给出Orlik–Solomon代数组合Leray模型的新解释。
The $OS$ algebra $A$ of a matroid $M$ is a graded algebra related to the Whitney homology of the lattice of flats of $M$. In case $M$ is the underlying matroid of a hyperplane arrangement \A in $\C^r$, $A$ is isomorphic to the cohomology algebra of the complement $\C^r\setminus \bigcup \A.$ Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic $OS$ algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic. We construct, for any given simple matroid $M_0$, a pair of infinite famili…
We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of $W$. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair $(W, W_L)$, where $W$ is arbitrary and $W_…
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to a component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in…
In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on the poset of layers. The analysis of discriminantal toric arrangements permits us to isolate certain conditions under which two toric arrangements have diffeomorphic complements. We also give combinatorial conditions determining whether the cohomology algebra is generated in degree one.
Cohomology ring of non-compact abelian arrangements
本文提出了一种非紧交换排列上同调环的Orlik-Solomon类型呈现，并通过比较不同交换群中的排列推导出所有其他交换排列的上同调环，同时给出了Orlik-Solomon和De Concini-Procesi关系的新证明。
Subarrangements of type A: the weak Lefschetz property of the Artinian Orlik-Terao algebra
本文研究了图排列的Artinian Orlik-Terao代数的弱Lefschetz性质（WLP），发现即使对于弦图（产生Koszul代数）的情况，WLP有时也会失效；而通过分析状态多面体可知，当所有可能的初始理想都失效时，WLP仍可能成立。更一般地，对于任何具有张量积分解的代数，我们构造了乘法映射的核中的标准元素，改进了文献中的先前结果。
We expand upon the notion of equivariant log concavity, and make equivariant log concavity conjectures for Orlik--Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik--Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{sl}_n$, we exploit the theory of representation stability to give computer assisted proofs of these conjectures in low degree.
In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of $W$, and gave a uniform proof of this claim for symmetric groups. In this note we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.
We study the cohomology ring of the complement $\mathcal{M}(\mathcal{A})$ of a manifold arrangement $\mathcal{A}$ in a smooth manifold $M$ without boundary. We first give the concept of monoidal cosheaf on a locally geometric poset $\mathfrak{L}$, and then define the generalized Orlik--Solomon algebra $A^*(\mathfrak{L}, \mathcal{C})$ over a commutative ring with unit, which is built by the classical Orlik--Solomon algebra and a monoidal cosheaf $\mathcal{C}$ as coefficients. Furthermore, we construct a monoidal cosheaf $\hat{\mathcal{C}}(\mathcal{A})$ associated with $\mathcal{A}$, so that the…
Let $\mathcal{A}$ be a Weyl arrangement in an $\ell$-dimensional Euclidean space. The freeness of restrictions of $\mathcal{A}$ was first settled by a case-by-case method by Orlik and the second author (1993), and later by a uniform argument by Douglass (1999). Prior to this, Orlik and Solomon (1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik, Solomon and the second author (1986), asserts that the exponents of any $A_1$ restriction, i.e., the restriction of $\mathcal{A}$ to a hyperplane, are given by $\{m_1,\ldots, m_{\ell-1}\}$…
Solomon--Terao Algebras: Top Degree, Dimension, and Homology
研究中心超平面排列的所罗门-寺尾代数及其复形，证明所罗门-寺尾代数的最高次数与一维最高分次分量的显式生成元，解决相关猜想，给出其总维数下界，还证明其与复形同调的 tame 性等价条件。
We prove that the ith graded pieces of the Orlik-Solomon algebras or Cordovil algebras of resonance arrangements form a finitely generated FS^op-module, thus obtaining information about the growth of their dimensions and restrictions on the irreducible representations of symmetric groups that they contain.
Associated to the cohomology ring A of the complement X(A) of a hyperplane arrangement A in complex m-space are the resonance varieties R^k(A). The most studied of these is R^1(A), which is the union of the tangent cones at the origin to the characteristic varieties of the fundamental group of X. R^1(A) may be described in terms of Fitting ideals, or as the locus where a certain Ext module is supported. Both these descriptions give obvious algorithms for computation. In this note, we show that interpreting R^1(A) as the locus of decomposable two-tensors in the Orlik-Solomon ideal leads to a de…
We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections. These results will then be used to characterize arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.
Resonance varieties and Lie algebras of matroids and hyperplane arrangements
本文综述拟阵 Orlik-Solomon 代数上的共振簇、完整李代数与完整 Chen 李代数，比较可实现与不可实现拟阵的异同并提出开放问题。
Let $\mathscr A$ be a Coxeter arrangement of rank $\ell$. In 1987 Orlik, Solomon and Terao conjectured that for every $1\leq d \leq \ell$, the first $d$ exponents of $\mathscr A$ -- when listed in increasing order -- are realized as the exponents of a free restriction of $\mathscr A$ to some intersection of reflecting hyperplanes of $\mathscr A$ of dimension $d$. This conjecture does follow from rather extensive case-by-case studies by Orlik and Terao from 1992 and 1993, where they show that all restrictions of Coxeter arrangements are free. We call a general free arrangements with this natura…
A net in $\mathbb{P}^2$ is a configuration of lines $\mathcal A$ and points $X$ satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac-Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid $M$ and rank $r$, we associate a monomial ideal (a monomial variant of the Orlik-Solomon ideal) to the set of flats of $M$ of rank $\le r$. In the context of line arrangements in $\mathbb{P}^2$, applying Alexander duality to the resulting ideal yields insight into the combinatorial str…
The Orlik-Solomon algebra A of a matroid is isomorphic to the quotient of an exterior algebra E by a defining ideal I. We find an explicit presentation of the annihilator ideal of I or, equivalently, the E-module dual to A. As an application of that we provide a necessary, combinatorial condition for the algebra A to be quadratic. We show that this is stronger than matroid being line-closed thereby resolving (negatively) a conjecture by Falk. We also show that our condition is not sufficient for the quadraticity.
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…
The cohomology ring of the boundary manifold of a combinatorial line arrangement
该研究计算组合直线排列的边界流形的整上同调环，将Cohen-Suciu加倍公式推广到含不可实现情形的任意组合直线排列，还推导了边界流形共振簇的相关结果。
Koszulity, supersolvability, and Stirling representations
本研究探究超可解超平面构形对应的Koszul代数与群作用的关联，以辫构形和对称群为例，揭示了Stirling数与代数希尔伯特函数的对应关系，还证明相关表示的分支规则与稳定性性质适用于所有超可解构形。
We examine the first non-vanishing higher homotopy group, $π_p$, of the complement of a hypersolvable, non--supersolvable, complex hyperplane arrangement, as a module over the group ring of the fundamental group, $\Zπ_1$. We give a presentation for the $I$--adic completion of $π_p$. We deduce that the second nilpotent $I$--adic quotient of $π_p$ is determined by the combinatorics of the arrangement, and we give a combinatorial formula for the second associated graded piece, $\gr^1_I π_p$. We relate the torsion of this graded piece to the dimensions of the minimal generating systems of the Orli…
Reconstruction of oriented matroids from Varchenko-Gelfand algebras
本文研究特征不等于2的系数环上的Varchenko-Gelfand代数，证明在余维2的一般超平面排列中，有向拟阵可从滤过和分次VG代数恢复，并给出一般情况下的重构算法。
Let $G$ be a matroid on ground set \A. The Orlik-Solomon algebra $A(G)$ is the quotient of the exterior algebra \E on \A by the ideal \I generated by circuit boundaries. The quadratic closure $\bar{A}(G)$ of $A(G)$ is the quotient of \E by the ideal generated by the degree-two component of \I. We introduce the notion of \nbb set in $G$, determined by a linear order on \A, and show that the corresponding monomials are linearly independent in the quadratic closure $\bar{A}(G)$. As a consequence, $A(G)$ is a quadratic algebra only if $G$ is line-closed. An example of S.~Yuzvinsky proves the conve…
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