open access

https://doi.org/10.14943/83521

https://hdl.handle.net/2115/69125

The double coxeter arrangement

英語

1997-04-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

375

1-21

Let V be Euclidean space. Let WC GL(V) be a finite irreducible reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H E A choose O'.H E V* such that H = ker( O'.H ). The arrangement A is known to be free: the derivation module D(A) = {8 E Ders I 8(aH) E SaH} is a free S-module with generators of degrees equal to the exponents of W. In this paper we prove an analogous theorem for the submodule E(A) of D(A) defined by E(A) = {8 E Ders I 8(aH) E Sa.1:,-}. The degrees of the basis elements are all equal to the Coxeter number. The module E(A) may be considered a deformation of the derivation module for the Shi arrangement, which is conjectured to be free. The proof is by explicit construction using a derivation introduced by K. Saito in his theory of flat generators.

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