Free hyperplane arrangements betweenAn−1 andBn
Free hyperplane arrangements betweenAn−1 andBn
复制标题
An−1 和 Bn 之间的自由超平面排列
DOI：
10.1007/bf02571719
复制
发表时间：
1994
影响因子：
0.8
通讯作者：
V. Reiner
中科院分区：
文献类型：
--
作者：
Paul H. Edelman;V. Reiner
超平面排列的研究动机主要来自于对Coxeter排列的研究。Coxeter安排的子安排不太好理解。在根系A，_ 1的情况下，子排列是图形排列，并且使用与图形的这种联系，可以从自由度的角度完全分析这些排列（例如定理3.3）。Zaslavsky开发了一个理论的签署图[Za]，在原则上，应该允许一个分析的子安排的B，以类似的方式。还没有人能够将其付诸实践。对于B的包含A，_I的子排列类，再次存在与图的自然连接。我们研究的就是这一类安排。在本文中，我们完全描述了这些安排是自由的，在一种类似的方式为子安排的情况下，A1_1-我们的部分动机是最近的工作J6 zefiak和Sagan [JS]对这些自由安排的某个子类。在下一节中，我们将收集有关排列的必要的已知结果，并建立图论术语。在W中，我们刻画了自由的A，_1的子排列，即那些由弦图参数化的子排列。这些结果大部分是已知的，但它们不在任何地方收集。在这一节中采用的技术也可以用来预示那些在w中，我们分类的子安排B，其中包含A，1和自由。它们由阈值图参数化。我们还表明，这些自由安排是超可解的。我们的分析的副产品是一个无限的集合的安排，是反例Orlik的猜想。此外，我们能够枚举从A，_~到B，.的所有归纳表。
Much of the motivation for the study of arrangements of hyperplanes comes from the study of Coxeter arrangements. The sub-arrangements of Coxeter arrangements are less well understood. In the case of the root system A, _ 1 the sub-arrangements are the graphic arrangements, and using this connection to graphs one can analyze these arrangements completely from the perspective of free-ness (eg Theorem 3.3). Zaslavsky has developed a theory of signed graphs [Za] which, in principle, should allow one to analyze the sub-arrangements of B, in a similar way. No one has yet been able to put that into practice. For the class of sub-arrangements of B, that contain A, _I there is again a natural connection to graphs. It is this class of arrangements that we study. In this paper we describe completely which of these arrangements are free in a manner analogous to the situation for sub-arrangements of A, _ 1-We were partly motivated by the recent work of J6zefiak and Sagan [JS] on a certain sub-class of these free arrangements.The paper is organized as follows. In the next section we collect the necessary known results concerning arrangements and establish graph-theoretic terminology. In w we characterize the sub-arrangements of A, _ 1 that are free, namely those parameterized by chordal graphs. For the most part these results are known, but they are not collected anywhere. The techniques employed in this section also serve to foreshadow those in w In w we classify the sub-arrangements of B, that contain A, _ 1 and are free. They are parameterized by threshold graphs. We also show which of these free arrangements are supersolvable. A by-product of our analysis is an infinite collection of arrangements that are counterexamples to Orlik's conjecture. In addition we are able to enumerate all of the induction tables from A, _~ to B,.
免费 Coxeter 安排和申请
DOI：
--
发表时间：
2010
期刊：
影响因子：
--
作者：
Taku Ishii;Miki Hirano;Tadashi Miyazaki;Shu Kawaguchi;石井卓・平野幹・宮崎直;平之内俊郎;Takuro Abe;鈴木正俊;山崎義徳;川口周;阿部拓郎;平之内俊郎;山崎義徳;鈴木正俊;Takuro Abe
通讯作者：
Takuro Abe