The Geometry of Root Systems and Signed Graphs
The Geometry of Root Systems and Signed Graphs
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DOI：
10.1080/00029890.1981.11995201
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发表时间：
1981-02
影响因子：
0.5
通讯作者：
T. Zaslavsky
中科院分区：
文献类型：
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作者：
T. Zaslavsky
这篇文章讲述了根系、图和拟阵之间的一种新发现的联系。根系统是满足一定对称性和度量正则性要求的向量集。它们出现在李论中，它们很重要，因为它们与李代数一一对应，因此也与李群一一对应，因为代数和群的许多性质都涉及到根系统。从那以后，他们发现了其他的应用，比如线形图和有限单群的搜索。图，或网络，由圆弧连接的节点组成，出现在各种组合分析中。然而，每个弧都用+或-标记的带符号图很少被讨论或应用。(以至于有些人一开始以为它们是另一种形式的有向图。事实并非如此。)它们将在本文中得到很好的应用，因为除了五个例外，每个根系统都可以简洁而忠实地用带符号的图表示。李论中遇到的一个问题是计算由根系统元素对偶的所有超平面切割成的空间的块数。通常的解决方法，这是经典的和众所周知的，依赖于将问题转化为一个关于根系统的自同构群的问题。但没有必要采取这种做法。相反，通过使用组合技术直接解决问题，人们不仅可以计算来自完整根系统的片段，还可以计算来自许多子系统的片段；粗略地说，根系统对应于完全图，而通过组合可解的附加系统对应于任意子图。主要的工具是超平面排列的特征多项式（见第4节），一个从矩阵理论借来的多项式。这个理论，我不需要再提它的名字，尽管如此，它仍然是我演讲的安静基础。
This essay tells of a newly discovered connection among root systems, graphs, and matroids. Root systems are sets of vectors which satisfy certain requirements of symmetry and metric regularity. They arose in Lie theory, where they are important because they correspond one-to-one to Lie algebras and hence to Lie groups and because many properties of the algebra and group involve the root system. They have since found other applications, such as to line graphs and the search for finite simple groups. 1 Graphs, or networks, which consist of nodes joined by arcs, arise in all kinds of combinatorial analysis. Yet signed graphs, in which each arc is labeled by+ or-, are rarely discussed or applied.(So much so that some people at first think they are another form of directed graph. They are not.) They will find good use in this article, for with five exceptions every root system can be concisely and faithfully represented by a signed graph. One of the problems encountered in Lie theory is that of counting the pieces into which space is cut by all the hyperplanes dual to elements of a root system. The usual method of solution, which is classical and well known, depends on translating the problem into one concerning an automorphism group of the root system. But it is not necessary to take that approach. Instead, by tackling the problem directly with combinatorial techniques, one can count the pieces derived not only from full root systems but from many subsystems; roughly speaking, the root systems correspond to complete graphs, while the additional systems solvable by combinatorics correspond to arbitrary subgraphs. The principal tool is the characteristic polynomial of an arrangement of hyperplanes (see Section 4), a polynomial borrowed from matroid theory. This theory, which I shall not need to mention again by name, is nonetheless the quiet ground of my discourse.