Abstract
In this article, we propose a generalization of the notion of chordal graphs to signed graphs, which is based on the existence of a perfect elimination ordering for a chordal graph. We give a special kind of filtrations of the generalized chordal graphs, and show a characterization of those graphs. Moreover, we also describe a relation between signed graphs and a certain class of multiarrangements of hyperplanes, and show a characterization of free multiarrangements in that class in terms of the generalized chordal graphs, which generalizes a well-known result by Stanley on free hyperplane arrangements. Finally, we give a remark on a relation of our results with a recent conjecture by Athanasiadis on freeness characterization for another class of hyperplane arrangements. Dans cet article, nous proposons une généralisation de la notion des graphes triangulés à graphes signés, qui est basée sur l'existence d'un ordre d'élimination simplicial à un graphe triangulé. Nous donnons un genre spécial de filtrations des graphes triangulés généralisés, et montrons une caractérisation de ces graphes. De plus, nous décrivons aussi une relation entre graphes signés et une certaine classe de multicompositions d'hyperplans, et montrons une caractérisation de multicompositions libres dans cette classe en termes des graphes triangulés généralisés, qui généralise un résultat célèbre de Stanley sur compositions libres d'hyperplans. Finalement, nous donnons une remarque sur une relation de nos résultats avec une conjecture récente d'Athanasiadis sur une caractérisation du freeness d'une autre classe de compositions d'hyperplans.
Highlights
Let V be an -dimensional vector space over a field K of characteristic zero
Braid arrangements, or more generally Coxeter arrangements, are fundamental objects in the theory of arrangements that are closely related to root systems of finite Coxeter groups (see e.g., Saito (1975))
0 otherwise, where vivj denotes an unordered pair of vi and vj, and put m = 2k + mG. (In this article, every graph is finite, simple, and undirected, unless otherwise specified.) One of the main theorems in this article gives a characterization of free multiplicities on A of the above type in terms of a certain property of the corresponding signed graph that will be described below
Summary
Let V be an -dimensional vector space over a field K of characteristic zero. A hyperplane arrangement A (or an arrangement) is a finite collection of affine hyperplanes in V. (In this article, every graph is finite, simple, and undirected, unless otherwise specified.) One of the main theorems in this article gives a characterization of free multiplicities on A of the above type in terms of a certain property of the corresponding signed graph that will be described below. From the viewpoint of Stanley’s freeness characterization based on (non-signed) chordal graphs, it is reasonable to expect that extending non-signed graphs to signed graphs gives a natural generalization of Stanley’s theory, and the corresponding multiplicities are of the above type Our result is applied to prove one direction of Athanasiadis’s Conjecture (the sufficiency of Athanasiadis’s conditions for the freeness) in a more general setting than that in the statement of the conjecture
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