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Multiderivations of Coxeter arrangements. (English) Zbl 1032.52013
Let $$V$$ be an $$\ell$$-dimensional Euclidean space, and $$S$$ the algebra of polynomial functions on $$V$$. Let $$\mathcal A$$ be a Coxeter arrangement, that is, the set of reflecting hyperplanes of a finite irreducible subgroup $$G$$ of the orthogonal group $$O(V)$$.
The paper studies the derivation modules $$D^{(m)}({\mathcal A})$$, introduced by G. M. Ziegler [Singularities, Proc. IMA Participating Inst. Conf., Iowa City/Iowa 1986, Contemp. Math. 90, 345-359 (1989; Zbl 0678.51010)]. The module $$D^{(m)}$$ consists of derivations $$\theta$$ on $$S$$ such that $$\theta(\alpha_H)\in S\alpha_H^m$$ for the defining linear form $$\alpha_H$$ of each hyperplane $$H\in {\mathcal A}$$. The author shows here that $$D^{(m)}$$ is a free $$S$$-module of rank $$\ell$$ for all $$m$$. The $$m=1$$ case was previously shown by K. Saito [On the uniformization of complements of discriminant loci. In: Conference Notes. Amer. Math. Soc. Summer Institute, Williamstown (1975)]. The $$m=2$$ case was shown by L. Solomon and H. Terao [Comment. Math. Helv. 73, 237-258 (1998; Zbl 0949.52009)]. The proof explicitly constructs a basis, and makes heavy use of the primitive derivation.

##### MSC:
 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 20F55 Reflection and Coxeter groups (group-theoretic aspects) 13N15 Derivations and commutative rings
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