Free arrangements of hyperplanes and unitary reflection groups
Free arrangements of hyperplanes and unitary reflection groups
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超平面和酉反射群的自由排列
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发表时间：
1980
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通讯作者：
H. Terao
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作者：
H. Terao
1、免费安排。我们将 C（或 P"/(C)）中的非空有限超平面族称为仿射（或射影）n 排列。如果 X 是仿射排列或射影 n 排列，则集合 X 简称为 n 排列。当 zex H# 时，n 排列 X 称为中心。用 IX] 表示 Jez H。令 X 为中心仿射 n 排列。通过对原点进行适当的翻译，我们可以假设 .ez H 包含 C"/ 中的原点 O。令 Q e C[zo, ..., z,] 为 IX[ 的无平方定义方程。 (C) 表示我们 C)c./,,o。则 D(X)’= {0;位于全纯向量场原点的胚芽，使得 0. Q e Q. 是一个模。如果 D(X) 是 ree 模，我们称 X 是自由的。假设中心仿射n-排列X是ree。令 {00, ., 0} 为 re 基系统或 D(X)，使得每个 0 都是 d 次齐次的。 （0 是齐次 o 度 d i 0 的表达式 o= ,f(/z), j=O，其中每个 f e C[Zo, ..., z,] 要么是 0，要么是齐次 o 度 d。）我们将整数 (do, ..., d) 称为 X 的广义指数。它们仅取决于 X [7]。令 X 为射影 n 排列。将 P"/(C) 简单地表示为 P"/。设 Q e C[zo, ., z,/] 是定义集合 [XIP/.那么存在唯一的中心仿射(n/1)-排列X使得v(e)-IXlC+，我们称X为自由i X为ree。假设投影 n 排列 X 是 ree。令(do, d, '', dn)为X的广义指数，那么我们可以假设do= 1（由于存在o2欧拉向量场
1. Free arrangements. We call a non-void finite amily of hyperplanes in C (or P"/(C)) an affine (resp. projective) n-arrangement. A set X is simply called an n-arrangemen if X is either an affine narrangement or a projective n-arrangement. An n-arrangement X is called to be central when zex H#. Denote Jez H by IX]. Let X be a central affine n-arrangement. By an appropriate translation of the origin we can assume that .ez H contains the origin O in C"/. Let Q e C[zo, ..., z,] be a square-free defining equation of IX[. By (C) denote we C)c./,,o. Then D(X)’= {0; a germ at the origin of holomorphie vector fields such that 0. Q e Q. is an -module. We call X to be free if D(X) is a ree -module. Assume that a central affine n-arrangement X is ree. Let {00, ., 0} be a system of ree basis or D(X) such that each 0 is homogeneous of degree d. (0 is homogeneous o degree d i 0 has an expression o= ,f(/z), j=O where each f e C[Zo, ..., z,] is either 0 or homogeneous o degree d.) We call the integers (do, ..., d) the generalized exponents of X. They depend only on X [7]. Let X be a projective n-arrangement. Denote P"/(C) simply by P"/. Let Q e C[zo, ., z,/] be a homogeneous polynomial defining a set [XI P/. Then there exists a unique central affine (n/ 1)-arrangement X such that v(e)-IXlC+, We cll X to be free i X is ree. Assume that a projective n-arrangement X is ree. Let (do, d, ’’, dn) be the generalized exponents of X, then we can assume that do= 1 (due to the existence o2 the Euler vector field