The Global Invariant of Signed Graphic hyperplane Arrangements
The Global Invariant of Signed Graphic hyperplane Arrangements
复制标题
有符号图形超平面排列的全局不变量
DOI：
10.1007/s00373-017-1770-y
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
G. Jiang
中科院分区：
文献类型：
--
作者：
Qiumin Guo;Weili Guo;Wentao Hu;G. Jiang
复向量空间中超平面排列的补的基本群是一个有趣而复杂的不变量。在基本群的较低中心级数中连续的三阶不变量被福尔克称为该排列的全局不变量。Falk给出了计算全局不变量的一般公式，并要求给出全局不变量的组合解释。Schenck和Suciu证明了一个图形排列的全局不变量是与该排列相关的图中具有三个或四个顶点的团的数量的两倍。这解决了福尔克的问题，在案件的图形安排。而在符号图的情形下，我们得到了一个类似的组合公式。本文给出了这个组合公式的一个直接而简单的证明。
The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an interesting and complicated invariant. The third rank of successive quotients in the lower central series of the fundamental group was called the global invariant of the arrangement by Falk. Falk gave a general formula to compute the global invariant, and asked for a combinatorial interpretation of the global invariant. Schenck and Suciu proved that the global invariant of a graphic arrangement is double of the number of cliques with three or four vertices in the graph with which the arrangement associated. This solved Falk’s problem in the case of graphic arrangements. While in the case of signed graphic arrangements, we obtained a similar combinatorial formula. In this paper, we give a direct and simple proof for this combinatorial formula.