Deformations of hypersolvable 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.

Procrustean statistical inference of deformations

A two step method has been devised for the statistical inference of deformation changes. In the first step of this method and based on Procrustes analysis of deformation tensors, the significance of the change in a time or space series of deformation tensors is statistically analyzed. In the second step significant change(s) in deformations are localized. In other words, they are assigned to ce...

Higher Homotopy Groups of Complements of Complex Hyperplane Arrangements

We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, and we give an explicit Zπ1-module presentation of πp, the first non-vanishing higher homotopy g...