Inventiones mathematicae

, Volume 148, Issue 3, pp 659–674 | Cite as

Multiderivations of Coxeter arrangements

  • Hiroaki Terao
Article

Abstract.

Let V be an ℓ-dimensional Euclidean space. Let GO(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 αHV* 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≤···≤m are the exponents of G and h=m+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.

Mathematical Subject Classification (2000): 32S22, 05E15, 20F55 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Hiroaki Terao
    • 1
  1. 1.Tokyo Metropolitan University, Mathematics Department, Minami-Ohsawa, Hachioji, Tokyo 192-0397, JapanJapan

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