Plus-one Generated and Next to Free Arrangements of Hyperplanes

Inductively Factored Signed-graphic Arrangements of Hyperplanes

In 1994, Edelman and Reiner characterized free and supersolvable hyperplane arrangements in the restricted interval [An−1, Bn]. In this paper, we give a characterization of inductively factored arrangements in this interval, and show that the same characterization also describes factored arrangements in this interval. These results use the compact notation of signed graphs introduced by Zaslavsky.

Division of characteristic polynomials and the division theorem for free arrangements of hyperplanes

We consider the triple (A,A′,AH) of hyperplane arrangements and the division of their characteristic polynomials. We show that the freeness of A and the division of χ(A; t) by χ(A ; t) confirm the freeness ofA. The key ingredient of this “division theorem” on freeness is the fact that, if χ(A ; t) divides χ(A; t), then the same holds for the localization at the codimension three flat in H. This...

Chambers of Arrangements of Hyperplanes and Arrow’s Impossibility Theorem

Let A be a nonempty real central arrangement of hyperplanes and Ch be the set of chambers of A. Each hyperplane H defines a half-space H and the other half-space H. Let B = {+,−}. For H ∈ A, define a map ǫ H : Ch → B by ǫ H (C) = + (if C ⊆ H) and ǫ H (C) = − (if C ⊆ H). Define ǫ H = −ǫ H . Let Ch = Ch×Ch× · · · ×Ch (m times). Then the maps ǫ H induce the maps ǫ H : Ch → B. We will study the adm...

Generalized Catalan Numbers, Weyl Groups and Arrangements of Hyperplanes

For an irreducible, crystallographic root system Φ in a Euclidean space V and a positive integer m, the arrangement of hyperplanes in V given by the affine equations (α, x)= k, for α∈Φ and k=0, 1, . . . ,m, is denoted here by AΦ . The characteristic polynomial of AΦ is related in the paper to that of the Coxeter arrangement AΦ (corresponding to m=0), and the number of regions into which the fun...

Arrangements of Hyperplanes and Vector Bundles on P