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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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