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.
In this paper, we study the class of free multiarrangements of hyperplanes. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over the rationals.
We will consider some characterizations of freeness of a hyperplane arrangement, in terms of the following properties: locally freeness, factorization of characteristic polynomial and freeness of restricted multiarrangement. In the case of 3-arrangement, freeness is characterized by factorization of characteristic polynomial and coincidence of its roots with exponents of restricted multiarrangement. In the case of higher dimension, it is characterized by a kind of locally freeness and freeness of restricted multiarrangement. As an application, we prove the freeness of certain arrangements whic…
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.
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.
Let A be a generic hyperplane arrangement composed of r hyperplanes in an n-dimensional vector space, and S the polynomial ring in n variables. We consider the S-submodule D(m)(A) of the nth Weyl algebra of homogeneous differential operators of order m preserving the defining ideal of A. We prove that if n \geq 3, r > n,m > r - n + 1, then D(m)(A) is free (Holm's conjecture). Combining this with some results by Holm, we see that D(m)(A) is free unless n \geq 3, r > n,m < r - n + 1. In the remaining case, we construct a minimal free resolution of D(m)(A) by generalizing Yuzvinsky's construction…
Let $W$ be a finite Weyl group and $\A$ be the corresponding Weyl arrangement. A deformation of $\A$ is an affine arrangement which is obtained by adding to each hyperplane $H\in\A$ several parallel translations of $H$ by the positive root (and its integer multiples) perpendicular to $H$. We say that a deformation is $W$-equivariant if the number of parallel hyperplanes of each hyperplane $H\in \A$ depends only on the $W$-orbit of $H$. We prove that the conings of the $W$-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers.…
Only few categories of free arrangements are known in which Terao's conjecture holds. One of such categories consists of $3$-arrangements with unbalanced Ziegler restrictions. In this paper, we generalize this result to arbitrary dimensional arrangements in terms of flags by introducing unbalanced multiarrangements. For that purpose, we generalize several freeness criterions for simple arrangements, including Yoshinaga's freeness criterion, to unbalanced multiarrangements.
Generalizing a result of Yoshinaga in dimension 3, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.
The freeness of hyperplane arrangements in a three dimensional vector space over finite field is discussed. We prove that if the number of hyperplanes is greater than some bound, then the freeness is determined by the characteristic polynomial.
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 $\mathbb{Q}$.
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.
In this paper we describe a parallel algorithm for generating all non-isomorphic rank $3$ simple matroids with a given multiplicity vector. We apply our implementation in the HPC version of GAP to generate all rank $3$ simple matroids with at most $14$ atoms and a splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank $3$ arrangement which is not inductively free has $14$ hyperplanes and exists in all characteristics di…
Let $A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $A''$ of $A$ to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of reflection arrangements. More precisely, let $A = A(W)$ be the reflection arrangement of a complex reflection group $W$. By work of Terao, each such reflection arrangement is free. Thus so is Ziegler's canonical multiplicity on the restriction…
An m-free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m-free imply (m+1)-free for any arrangement? (2)Are all arrangements m-free for m large enough? In this paper, we characterize m-freeness for product arrangements, while we prove that all localizations of an m-free arrangement are m-free. From these results, we give answers to Holm's questions.
A Weyl arrangement is the hyperplane arrangement defined by a root system. Arnold and Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type $A_{\ell}$ are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of thier graph. In addition, The Weyl subarrangements of type $B_{\ell}$ can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness …
Let $ G $ be a simple graph of $ \ell $ vertices $ \{1, \dots, \ell \} $ with edge set $ E_{G} $. The graphical arrangement $ \mathcal{A}_{G} $ consists of hyperplanes $ \{x_{i}-x_{j}=0\} $, where $ \{i, j \} \in E_{G} $. It is well known that three properties, chordality of $ G $, supersolvability of $ \mathcal{A}_{G} $, and freeness of $ \mathcal{A}_{G} $ are equivalent. Recently, Richard P. Stanley introduced $ ψ$-graphical arrangement $ \mathcal{A}_{G, ψ} $ as a generalization of graphical arrangements. Lili Mu and Stanley characterized the supersolvability of the $ ψ$-graphical arrangemen…
We prove a characterization of freeness, conjectured by Athanasiadis, for the family of hyperplane arrangements which lie between the Coxeter and the Catalan arrangement of type $A_\ell$. One direction was already proved in [2]. Here we prove the other direction
We consider the triple $(\mathcal{A},\mathcal{A}',\mathcal{A}^H)$ of hyperplane arrangements and the division of their characteristic polynomials. We show that the freeness of $\mathcal{A}^H$ and the division of $χ(\mathcal{A};t)$ by $χ(\mathcal{A}^H;t)$ confirm the freeness of $\mathcal{A}$. The key ingredient of this "division theorem" on freeness is the fact that, if $χ(\mathcal{A}^H;t)$ divides $χ(\mathcal{A};t)$, then the same holds for the localization at the codimension three flat in $H$. This implies the local-freeness of $\mathcal{A}$ in codimension three along $H$. Based on these res…
Alon and Füredi (European J. Combin., 1993) proved that any family of hyperplanes that covers every point of the Boolean cube $\{0,1\}^n$ except one must contain at least $n$ hyperplanes. We obtain two extensions of this result, in characteristic zero, for hyperplane covers of symmetric sets of the Boolean cube (subsets that are closed under permutations of coordinates), as well as for `polynomial covers' of `weight-determined' sets of `strictly unimodal uniform' (SU$^2$) grids. As a main tool for solving our problems, we give a combinatorial characterization of (finite-degree) Zariski (Z-) cl…
We will prove the freeness of multi-Coxeter arrangements by constructing a basis of the module of vector fields which contact to each reflecting hyperplanes with some multiplicities using K. Saito's theory of primitive derivation.
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…
Call a function f : F_2^n -> {0,1} odd-cycle-free if there are no x_1, ..., x_k in F_2^n with k an odd integer such that f(x_1) = ... = f(x_k) = 1 and x_1 + ... + x_k = 0. We show that one can distinguish odd-cycle-free functions from those eps-far from being odd-cycle-free by making poly(1/eps) queries to an evaluation oracle. To obtain this result, we use connections between basic Fourier analysis and spectral graph theory to show that one can reduce testing odd-cycle-freeness of Boolean functions to testing bipartiteness of dense graphs. Our work forms part of a recent sequence of works tha…
We establish the relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space, respectively projective space associated to the module of logarithmic vector fields along the arrangement. Our main result gives a Künneth formula connecting the cohomology theories, answering a question by Yoshinaga. This, in turn, provides a characterization of the projective dimension of the module of logarithmic vector fields and yields a new proof of Yuzvinsky's freeness c…
Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note, we prove that arrangements that admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end, we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary, we give…
In this paper, we show how to compute using Fitting ideals the nonfree locus of the moduli space of arrangements of a rank $3$ simple matroid, i.e., the subset of all points of the moduli space which parametrize nonfree arrangements. Our approach relies on the so-called Ziegler restriction and Yoshinaga's freeness criterion for multiarrangements. We use these computations to verify Terao's freeness conjecture for rank $3$ central arrangements with up to $14$ hyperplanes in any characteristic.
We introduce a new class of arrangements of hyperplanes, called (strictly) plus-one generated arrangements, from algebraic point of view. Plus-one generatedness is close to freeness, i.e., plus-one generated arrangements have their logarithmic derivation modules generated by dimension plus one elements, with relations containing one linear form coefficient. We show that strictly plus-one generated arrangements can be obtained if we delete a hyperplane from free arrangements. We show a relative freeness criterion in terms of plus-one generatedness. In particular, for plane arrangements, we show…
Let ${\mathcal C}= \bigcup_{i=1}^n C_i \subseteq \mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a large class of arrangements of smooth rational curves, giving an inductive tool for understanding the freeness of the module $Ω^1({\mathcal C})$ of logarithmic differential forms with pole along ${\mathcal C}$. We also show that the analog of Terao's conjecture (freeness of $Ω^1({\mathcal C})$ is combinatorially determined if ${\mathcal C}$ is a union of lines…
We describe $Forb\{K_{1,3}, \overline {K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $ \overline{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths or isolated vertices, and of the complements of these graphs. Considering the hypergraph ${\mathcal H} ^{(3)}(G)$ made of the 3-element subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an independent subset, we deduce from above a description of the Boolean sum $G\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hy…
We describe ${\rm Forb}\{K_{1,3}, \bar {K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $\bar{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths, and of the complements of these graphs. Considering the hypergraph ${\mathcal H}^{(3)}(G)$ made of the 3-element subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an independent subset, we deduce from above a description of the Boolean sum $G\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hypergraph. We indicate the r…
In the study of free arrangements, the most useful result to construct/check free arrangements is the addition-deletion theorem. Recently, the multiple version of the addition theorem is proved, called the multiple addition theorem (MAT) to prove the ideal-free theorem. The aim of this article is to give the deletion version of MAT, the multiple deletion theorem (MDT). Also, we can generalize MAT from the viewpoint of our new proof. Moreover, we introduce their restriction version, a multiple restriction theorem (MRT). Applications of them including the combinatorial freeness of the extended C…
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…
Motivated by some computations of Feynman integrals and certain conjectures on mixed Tate motives, Bejleri and Marcolli posed questions about the $\mathbb{F}_1$-structure (in the sense of torification) on the complement of a hyperplane arrangement, especially for an arrangement defined in the space of cycles of a graph. In this paper, we prove that an arrangement has an $\mathbb{F}_1$-structure if and only if it is Boolean. We also prove that the arrangement in the cycle space of a graph is Boolean if and only if the cycle space has a basis consisting of cycles such that any two of them do not…
We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. This gives a lot of corollaries including the existence of free filtrations. The proof is based on the result about the form of minimal generators of a logarithmic derivation module of a multiarrangement which satisfies the $b_2$-equality.
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study logarithmic derivations and critical set varieties of arrangements in a way which is symmetric with respect to matroid duality. Our main result exhibits the variety of the ideal of pairs as a subspace arrangement whose components correspond to cyclic flats of the arrangement. As a corollary, we are able to give geometric explanations of some freeness and projec…
In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally …
The reflection arrangement of a Coxeter group is a well known instance of a free hyperplane arrangement. In 2002, Terao showed that equipped with a constant multiplicity each such reflection arrangement gives rise to a free multiarrangement. In this note we show that this multiarrangment satisfies the stronger property of inductive freeness in case the Coxeter group is of type $A$.
We study a family of symmetric polynomials that we refer to as the Boolean product polynomials. The motivation for studying these polynomials stems from the computation of the characteristic polynomial of the real matroid spanned by the nonzero vectors in $\mathbb{R}^n$ all of whose coordinates are either $0$ or $1$. To this end, one approach is to compute the zeros of the Boolean product polynomials over finite fields. The zero loci of these polynomials cut out hyperplane arrangements known as resonance arrangements, which show up in the context of double Hurwitz polynomials. By relating the …
We propose an approach to study logarithmic sheaves T(-log A) associated with a hyperplane arrangements A on the projective space, based on projective duality, direct image functors and vector bundles methods. We focus on freeness of line arrangements having a point with high multiplicity.
In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. We then extend our results further to all imprimitive irreducible unitary reflection groups. In this case the exponents turn out to depend on the expon…
Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A = A(W) be the associated hyperplane arrangement of W. Terao has shown that each such reflection arrangement A is free. There is the stronger notion of an inductively free arrangement. In 1992, Orlik and Terao conjectured that each reflection arrangement is inductively free. It has been known for quite some time that the braid arrangement as well as the Coxeter arrangements of type B and type D are inductively free. Barakat and Cuntz completed this list only recently by showing that every Coxeter ar…
In [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is pu…
The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, …
The addition-deletion theorems for hyperplane arrangements, which were originally shown in [H. Terao, Arrangements of hyperplanes and their freeness I, II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293--320], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition-deletion theorems to various a…
