Coxeter and crystallographic arrangements are inductively free

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...

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...

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...

Partial Normalizations of Coxeter Arrangements and Discriminants

We study natural partial normalization spaces of Coxeter arrangements and discriminants and relate their geometry to representation theory. The underlying ring structures arise from Dubrovin’s Frobenius manifold structure, which is lifted (without unit), to the space of the arrangement. We also describe an independent approach to these structures via duality of maximal Cohen–Macaulay fractional...