Characteristic Polynomials of Subspace Arrangements and Finite Fields

Athanasiadis, Christos A.

doi:10.1006/aima.1996.0059

Cited by 153 publications

(274 citation statements)

References 27 publications

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“…Now let us go back to the deformation (5.1). A useful way to consider this arrangement is introduced by Athanasiadis in [7]. Consider the directed graph G consisting of the vertex set V G = {1, 2, .…”

Section: Conjecture Of Athanasiadismentioning

confidence: 99%

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Signed-eliminable graphs and free multiplicities on the braid arrangement

Abe

Nuida²,

Numata³

2009

Journal of the London Mathematical Society

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Abstract. We define specific multiplicities on the braid arrangement by using signed graphs. To consider their freeness, we introduce the notion of signedeliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of certain deformations of the braid arrangement in terms of directed graphs. IntroductionLet V = V ℓ be an ℓ-dimensional vector space over a field K of characteristic zero, {x 1 , . . . , x ℓ } a basis for the dual vector space V * and S := Sym(Vis homogeneous of degree p if f i is zero or homogeneous of degree p for each i.A hyperplane arrangement A (or simply an arrangement ) is a finite collection of affine hyperplanes in V . If each hyperplane in A contains the origin, we say that A is central. In this article we assume that all arrangements are central unless otherwise specified. A multiplicity m on an arrangement A is a map m : A → Z ≥0 and a pair (A, m) is called a multiarrangement. Let |m| denote the sum of the multiplicities H∈A m(H). When m ≡ 1, (A, m) is the same as the hyperplane arrangement A and sometimes called a simple arrangement. For each hyperplane H ∈ A fix a linear form α H ∈ V * such that ker(α H ) = H. The first main object in this article is the logarithmic derivation module D(A, m) of (A, m) defined byis free, then there exists a homogeneous free basis {θ 1 , . . . , θ ℓ } for D(A, m). Then we define the exponents of a free multiarrangement (A, m) by exp(A, m) := (deg(θ 1 ), . . . , deg(θ ℓ )). The exponents are independent of a choice of a basis. When m ≡ 1, the logarithmic derivation module and exponents are denoted by D(A) and exp(A). When we fix a simple arrangement A, we say that a multiplicity m on A is free (resp. non-free) if a multiarrangement (A, m) is free (resp. non-free).A fundamental object of study in hyperplane arrangements is the arrangement of all reflecting hyperplanes of a Coxeter group, called a Coxeter arrangement. [26]. In this article we generalize the study of free multiplicities on the braid arrangement.A braid arrangement A ℓ , or the Coxeter arrangement of type A ℓ is defined as. By using the primitive derivation introduced in [16], free multiplicities on Coxeter arrangements are studied by , Terao [21], Yoshinaga [23], and the first author and Yoshinaga [6]. Combining these results, we have a characterization of the freeness of quasi-constant multiplicities m on a Coxeter arrangement, i.e., multiplicities such that max H,H ′ ∈A |m(H) − m(H ′ )| ≤ 1. However, it is known that if max H,H ′ ∈A |m(H) − m(H ′ )| = 2 then the same method using the primitive derivation does not work. Also, to determine explicitly which multiplicity makes (A, m) free is a difficult problem. Our aim is to consider these multiplicities on the braid arrangement and classify their freeness completely. In fact...

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Section: Conjecture Of Athanasiadismentioning

confidence: 99%

“…For such a graph G let A G denote the corresponding arrangement of the form (5.1). In [7], Athanasiadis gave a splitting formula of the characteristic polynomial of A G when G satisfies the following two conditions:…”

Section: Conjecture Of Athanasiadismentioning

confidence: 99%

Signed-eliminable graphs and free multiplicities on the braid arrangement

Abe

Nuida²,

Numata³

2009

Journal of the London Mathematical Society

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“…Several authors have worked on computing the number of regions of specific hyperplane arrangements. See for example [1,2,7].…”

Section: Introductionmentioning

confidence: 99%

“…By finite field methods (see [2,3,10]), one can solve these problems without much difficulty. But Stanley asked for combinatorial proofs, i.e., bijections between the regions of the hyperplane arrangements and objects that are counted by the corresponding formulas.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Enumeration of the Regions of Some Linear Arrangements

Seo¹

2016

Bulletin of the Korean Mathematical Society

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“…These facts provide the "finite field method" to study the real arrangement A. The method was initiated and systematically applied by Athanasiadis [1,2,3]. It has been used to solve problems related to hyperplane arrangements by Björner and Ekedahl [7] and Blass and Sagan [8] among others.…”

Section: Introductionmentioning

confidence: 99%

Periodicity of hyperplane arrangements with integral coefficients modulo positive integers

2007

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We study central hyperplane arrangements with integral coefficients modulo positive integers q. We prove that the cardinality of the complement of the hyperplanes is a quasi-polynomial in two ways, first via the theory of elementary divisors and then via the theory of the Ehrhart quasi-polynomials. This result is useful for determining the characteristic polynomial of the corresponding real arrangement. With the former approach, we also prove that intersection lattices modulo q are periodic except for a finite number of q's.

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Characteristic Polynomials of Subspace Arrangements and Finite Fields

Cited by 153 publications

References 27 publications

Signed-eliminable graphs and free multiplicities on the braid arrangement

Signed-eliminable graphs and free multiplicities on the braid arrangement

Combinatorial Enumeration of the Regions of Some Linear Arrangements

Periodicity of hyperplane arrangements with integral coefficients modulo positive integers

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