Signed-eliminable graphs and free multiplicities on the braid arrangement

Abstract

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...

A characterization of signed graphs with generalized perfect elimination orderings

“…It is also worthy to notice that Zaslavsky's work did not present a generalization of chordality to signed graphs (see [12,Section 5.2]), while our characterization (Theorem 5.1) shows a generalization of chordality that is compatible to the generalized perfect elimination orderings. Moreover, our above-mentioned work [2] that uses the result of the present paper is also relevant to a conjecture proposed by Athanasiadis [3] on the freeness characterization for another class of hyperplane arrangements; one direction of the conjecture (namely, implication of freeness from the conditions proposed by Athanasiadis) is proven in [2, Section 5] as an application of our results. These would show the significance of the generalization of perfect elimination orderings proposed in the present paper.…”

“…Recently, Abe, Numata and the author [2] would like to extend this equivalence to Coxeter arrangements of type A n that are endowed with multiplicities of some types. Our multiplicities are naturally parameterized by signed graphs in such a way that pairs of non-adjacent vertices correspond to ''neutral'' multiplicities 2m, and plus-and minus-signed edges correspond to 2m + 1 and 2m − 1, respectively (the case m = 0 corresponds to the original case of sub-arrangements of type A n ).…”

“…Thus we extended a part of Edelman and Reiner's result to the case of multiarrangements by virtue of the result of the present paper. See [2] for more details.…”

Totally free arrangements of hyperplanes

“…Recently the first and second authors and Wakefield developed a general theory of free multiarrangements and introduced the concept of free multiplicity in [3] and [4]. Several papers including [1], [2], [5] and [11] studied the set of free multiplicities for a fixed arrangement A. The main theorem (Theorem 1.2) in this article shows that the set of free multiplicities (or N F M(A)) imposes strong restrictions on the original arrangement A.…”

On the Extendability of Free Multiarrangements

“…Recently Abe, Nuida and Numata obtained more general results for type A ℓ arrangements [2,9]. Their results suggest that (5.1) holds even for wider class of multiplicities, namely, m : A → {−1, 0, 1}.…”