您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

山东大学学报（理学版） ›› 2018, Vol. 53 ›› Issue (6): 70-75.doi: 10.6040/j.issn.1671-9352.0.2017.644

• • 上一篇    下一篇

Weyl构形An-1和Bn之间的构形的自由性

高瑞梅,初颖*   

  1. 长春理工大学理学院, 吉林 长春 130022
  • 收稿日期:2017-12-19 出版日期:2018-06-20 发布日期:2018-06-13
  • 作者简介:高瑞梅(1983— ), 女, 博士, 副教授, 研究方向为奇点理论和超平面构形. E-mail:gaorm135@nenu.edu.cn*通信作者简介:初颖(1984— ), 女, 博士, 讲师, 研究方向为奇异椭圆方程的求解问题. E-mail:chuying_12345@sina.com
  • 基金资助:
    国家自然科学基金资助项目(11501051);吉林省科技发展计划项目优秀青年人才基金项目(20180520025JH)

Freeness of arrangements between the Weyl arrangements of types An-1 and Bn

GAO Rui-mei, CHU Ying*   

  1. Department of Science, Changchun University of Science and Technology, Changchun 130022, Jilin, China
  • Received:2017-12-19 Online:2018-06-20 Published:2018-06-13

PDF (PC)

436

摘要： Weyl群的反射超平面形成的集合称为该群对应的Weyl构形。设符号An-1和Bn分别表示An-1型和Bn型Weyl构形。若构形A满足An-1⊂A⊂Bn, 则称A为An-1和Bn之间的构形。本文首先研究了阈图,给出构造阈图的一种方法。 然后,利用阈图研究了An-1和Bn之间的构形的自由性。给出结论:对于满足|An-1|n|的任意整数k, 均存在An-1和Bn之间的自由构形, 其基数为k。同时也给出An-1和Bn之间的非自由构形的类似的结论。

关键词: 图构形, 自由性, Weyl构形, 阈图, 符号图

Abstract: The set of reflecting hyperplanes of a Weyl group is called Weyl arrangement. Assume the notations An-1 and Bn denote the Weyl arrangements of types An-1 and Bn respectively. The arrangement A which satisfies An-1⊂A⊂Bn is called the arrangement between An-1 and Bn. Firstly, we study threshold graphs, and give a construction for threshold graphs. Secondly, we study the freeness of the arrangements between An-1 and Bn by using threshold graphs. And we conclude: for any integer k satisfying |An-1|n|, there exists a free arrangement between An-1 and Bn with cardinality k. A similar conclusion for the non-freeness arrangements between An-1 and Bn is given.

Key words: Weyl arrangement, threshold graph, signed graph, graphical arrangement, freeness

中图分类号: 

  • O189
[1] SAITO K. Theory of logarithmic differential forms and logarithmic vector fields[J]. Journal of the Faculty of Science, the University of Tokyo, Section IA, Mathematics, 1980, 27(2):265-291.
[2] ABE T, BARAKAT M, CUNTZ M, et al. The freeness of ideal subarrangements of Weyl arrangements[J]. Journal of the European Mathematical Society, 2016, 18(6):1339-1348.
[3] ABE T. Divisionally free arrangements of hyperplanes[J]. Inventiones Mathematicae, 2016, 204(1):317-346.
[4] ABE T, TERAO H. Simple-root bases for Shi arrangements[J]. Journal of Algebra, 2015, 422(6):89-104.
[5] 高瑞梅. G2型Shi-Catalan构形的自由性[J]. 山东大学学报(理学版), 2014, 49(12):66-70+94. GAO Ruimei. The freeness of Shi-Catalan arrangements of type G2[J]. Journal of Shandong University(Natural Science), 2014, 49(12):66-70+94.
[6] GAO Ruimei, PEI Donghe, TERAO H. The Shi arrangement of the type Dl[J]. Proceedings of the Japan Academy, Mathematical Sciences(Series A), 2012, 88(3):41-45.
[7] STANLEY R P. An introduction to hyperplane arrangements[J]. In Geometric Combinatorics, IAS/Park City Math Ser, American Mathematical Society, Providence, RI, 2007, 13:389-496.
[8] EDELMAN P H, REINER V. Free hyperplane arrangements between An-1 and Bn[J]. Mathematische Zeitschrift, 1994, 215(1):347-365.
[9] ORLIK P, TERAO H. Arrangements of hyperplanes[M] // Grundlehren der Mathematischen Wissenschaften. Berlin: Springer-Verlag, 1992: 1-325.
[10] ZASLAVSKY T. Signed graph[J]. Discrete Applied Mathematics, 1982, 4(1):47-74.
[11] CHVÁTAL V, HAMMER P L. Aggregation of inequalities in integer programming[J]. Annals of Discrete Mathematics, 1977, 32(1):145-162.
[12] JIANG Guangfeng, YU Jianming. Supersolvability of complementary signed-graphic hyperplane arrangements[J]. Australasian Journal of Combinatorics, 2004, 30:261-276.
[13] GOLUMBIC M C. Trivially perfect graphs[J]. Discrete Mathematics, 1978, 24(1):105-107.
[1] 高瑞梅,李喆. 简单相连多边形对应的图构形的特征多项式[J]. 山东大学学报（理学版）, 2016, 51(10): 72-77.
[2] 高瑞梅. G2型Shi-Catalan构形的自由性[J]. 山东大学学报（理学版）, 2014, 49(12): 66-70.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发