Multiderivations of Coxeter arrangements

By Hiroaki Terao. 2000

70 citations recorded by the source corpus.

Abstract.Let V be an ℓ-dimensional Euclidean space. Let G⊂O(V) be a finite irreducible orthogonal reflection group. Let ? be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H∈? choose αH∈V* such that H=ker(αH). For each nonnegative integer m, define the derivation module D(m)(?)={θ∈DerS|θ(αH)∈SαmH}. The module is known to be a free S-module of rank ℓ by K. Saito (1975) for m=1 and L. Solomon-H. Terao (1998) for m=2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D(m)(?). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m−1)h/2)+mi(1≤i≤ℓ) (when m is odd). Here m1≤···≤mℓ are the exponents of G and h=mℓ+1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G). Some new results concerning the primitive derivation D are obtained in the course of proof of the main result.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b3be1a21fa64558f8cd3c51293828636a30dfa2e

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Citations in the corpus

The Euler multiplicity and addition–deletion theorems for multiarrangements (2006, unknown): “m1 1 (x1 −x3) m2(x 1 −x2) m3xm4 2 (x2 −x3) m5xm6 3 . The ﬁltration is given by A1 : = {x1 = 0}, A2 : = {x1x2(x1 −x2) = 0}, A3 : = {x1x2x3(x1 −x2)(x1 −x3)(x2 −x3) = 0}. It is shown in [Sa1], [Sa2] and [T4] that m= [m1 ...,m6] = [m,m,m,m,m,m] for m∈ Z>0 is a free multiplicity of A. Now we obtain th”
THE CHARACTERISTIC POLYNOMIAL OF A MULTIARRANGEMENT (2006, background): “Other examples of free multiarrangements include the restricted multiarrangements of a free arrangement (see [17]) and the Coxeter arrangements with a constant multiplicity (see [11] and [13]).”
Freeness of hyperplane arrangements and related topics (2012, background): “([5, 35, 41, 42, 46]) Let A be a Coxeter arrangement with exponents (e1, .”
Characterization of a free arrangement and conjecture of Edelman and Reiner (2004, background): “In this direction, the freeness of multi Coxeter arrangement which plays a crucial role in this paper has been proved by Terao [ Te3 ] (Theorem 2.5 below).”