| Reducibility of hyperplane arrangements |
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| 作者单位： | Guang-feng JIANG（Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China） ;
Jian-ming YU（Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China） ; |
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| 摘 要： | Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are.
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| 收稿时间： | 4 February 2006 |
| 修稿时间： | 1 August 2006 |
Reducibility of hyperplane arrangements |
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| Authors: | Guang-feng Jiang Jian-ming Yu |
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| Institution: | 1. Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China
2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China |
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| Abstract: | Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are. |
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| Keywords: | hyperplane arrangement irreducible component logarithmic derivation |
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