Deformations of Coxeter Hyperplane Arrangements

Deformations of Coxeter Hyperplane Arrangements

Deformations of Coxeter Hyperplane Arrangements

We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement xi − xj = 1, 1 ≤ i < j ≤ n, is equal to the number of alternating trees on n + 1 vertices. Remarkably, these numbers have several additional combinatorial interpretations in terms of bi...

Deformations of Coxeter hyperplane arrangements and their characteristic polynomials

Let A be a Coxeter hyperplane arrangement, that is the arrangement of reflecting hyperplanes of an irreducible finite Coxeter group. A deformation of A is an affine arrangement each of whose hyperplanes is parallel to some hyperplane of A. We survey some of the interesting combinatorics of classes of such arrangements, reflected in their characteristic polynomials.

Multiderivations of Coxeter arrangements

Let V be an l-dimensional Euclidean space. Let G ⊂ O(V ) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the derivation module D(A) = {θ ∈ DerS | θ(αH) ∈ SαmH}. The module is known to be a free S-module of...

Hodge Spectrum of Hyperplane Arrangements

In this article there are two main results. The first result gives a formula, in terms of a log resolution, for the graded pieces of the Hodge filtration on the cohomology of a unitary local system of rank one on the complement of an arbitrary divisor in a smooth projective complex variety. The second result is an application of the first. We give a combinatorial formula for the spectrum of a h...