The double Coxeter arrangement
The double Coxeter arrangement
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双考克斯特排列
DOI：
10.1007/s000140050054
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发表时间：
1997
影响因子：
0.9
通讯作者：
H. Terao
中科院分区：
文献类型：
--
作者：
Louis Solomon;H. Terao
Abstract. Let V be Euclidean space. Let
$ W \subset {\bf G L}(V) $ be a finite irreducible reflection group. Let
$ \cal {A} $ be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For
$ H \in \cal {A} $ choose
$ \alpha_H \in V^* $ such that
$ H = {\rm ker}(\alpha_H) $. The arrangement
$ {\cal A} $ is known to be free: the derivation module
$ D({\cal A}) = \{ \theta \in {\rm Der}_S ~|~ \theta(\alpha_H) \in S \alpha_H \} $ 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({\cal A}) $ of
$ D({\cal A})$ defined by
$ E({\cal A}) = \{\theta \in {\rm Der}_S ~|~ \theta(\alpha_H) \in S \alpha_H^2\} $. The degrees of the basis elements are all equal to the Coxeter number. The module
$ E({\cal 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.
Abstract. Let V be Euclidean space. Let
$ W \subset {\bf G L}(V) $ be a finite irreducible reflection group. Let
$ \cal {A} $ be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For
$ H \in \cal {A} $ choose
$ \alpha_H \in V^* $ such that
$ H = {\rm ker}(\alpha_H) $. The arrangement
$ {\cal A} $ is known to be free: the derivation module
$ D({\cal A}) = \{ \theta \in {\rm Der}_S ~|~ \theta(\alpha_H) \in S \alpha_H \} $ 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({\cal A}) $ of
$ D({\cal A})$ defined by
$ E({\cal A}) = \{\theta \in {\rm Der}_S ~|~ \theta(\alpha_H) \in S \alpha_H^2\} $. The degrees of the basis elements are all equal to the Coxeter number. The module
$ E({\cal 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.
DOI：
10.1073/pnas.93.6.2620
发表时间：
1996-03
影响因子：
11.1
作者：
R. Stanley
通讯作者：
R. Stanley