Coxeter arrangements in three dimensions

Coxeter Arrangements in Three Dimensions

Let A be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of A are isometric. We prove that A is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open questio...

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...

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...

Random Walks and Plane Arrangements in Three Dimensions

This paper explains some modern geometry and probability in the course of solving a random walk problem. Consider n planes through the origin in three dimensional Euclidean space. Assume, for simplicity, that they are in \general position". They then divide space into n(n ?1)+2 regions. We study a random walk on these regions. Suppose the walk is in region C. Pick a pair of the planes at random...