有限反射群（Coxeter群）的最长元的一些问题
|
摘要
本学位论文研究了有限反射群,分为三章.
第一章,主要介绍了本论文的一些概念。我们介绍了反射群和Weyl群的相关知识,并指出了有限Coxeter群与Weyl群之间的关系。最后我们还介绍了Coxeter群的抛物子群和cuspidal类的相关知识。
第二章,主要研究有限Coxeter群的最长元的一般表达式。我们还发现典型的Weyl群的最长元的表达式的规律,并通过推广得到了它们的一般表达式。
第三章,研究了有限Coxeter群的最长元的共轭类与cuspidal类。首先我们给出了最长元的共轭类是cuspidal类的有限Coxeter群,也给出了最长元的共轭类不是cuspidal类的有限Coxeter群。最后刻画了有限Coxeter群内与最长元的共轭的长度最短的元素的集合以及该集合中元素的个数。
In this thesis, some finite reflection groups are studied. The thesis is divided into three chapters.
In Chapter 1, we introduce the finite reflection groups and Weyl groups. Furthermore, we point out the relation between Coxeter groups and Weyl groups. Lastly, We introduce the parabolic subgroups of Coxeter groups and cuspidal class.
In Chapter 2, we investigate the general expression of the longest element in finite reflection Coxeter groups. We find out the rule in the expression of the longest element in the classical Weyl groups, and give their general expressions.
The aim of Chapter 3 is to study the conjugacy class and cuspidal class of the longest element in the finite Coxeter groups. We figure out some Coxeter groups whose conjugacy class of the longest element are cuspidal class and which are not cuspidal class. In addition, we also describe the set of elements of minimal length in conjugacy class of the longest elements in the Coxeter groups and give their the orders.
第一章,主要介绍了本论文的一些概念。我们介绍了反射群和Weyl群的相关知识,并指出了有限Coxeter群与Weyl群之间的关系。最后我们还介绍了Coxeter群的抛物子群和cuspidal类的相关知识。
第二章,主要研究有限Coxeter群的最长元的一般表达式。我们还发现典型的Weyl群的最长元的表达式的规律,并通过推广得到了它们的一般表达式。
第三章,研究了有限Coxeter群的最长元的共轭类与cuspidal类。首先我们给出了最长元的共轭类是cuspidal类的有限Coxeter群,也给出了最长元的共轭类不是cuspidal类的有限Coxeter群。最后刻画了有限Coxeter群内与最长元的共轭的长度最短的元素的集合以及该集合中元素的个数。
In this thesis, some finite reflection groups are studied. The thesis is divided into three chapters.
In Chapter 1, we introduce the finite reflection groups and Weyl groups. Furthermore, we point out the relation between Coxeter groups and Weyl groups. Lastly, We introduce the parabolic subgroups of Coxeter groups and cuspidal class.
In Chapter 2, we investigate the general expression of the longest element in finite reflection Coxeter groups. We find out the rule in the expression of the longest element in the classical Weyl groups, and give their general expressions.
The aim of Chapter 3 is to study the conjugacy class and cuspidal class of the longest element in the finite Coxeter groups. We figure out some Coxeter groups whose conjugacy class of the longest element are cuspidal class and which are not cuspidal class. In addition, we also describe the set of elements of minimal length in conjugacy class of the longest elements in the Coxeter groups and give their the orders.
引文
[1] James E. Humphreys. The Reflection Groups and Coxeter Groups, Cambridge university press,1990.
[2] R.W.Richardson. Conjugacy classes of involutions in Coxeter groups,Bull. Austral. Math. Soc. 26(1982), 1-15. (68,101).
[3] Meinolf Geck, GOtz Pfeiffer. Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, Clarendon press . oxford,2000.
[4] Gong-qiang Wang, Xi-gou Zhang, Chun Liu. A Special of XiangNan University, 2009(2).
[5] Pu Yanmin, Chen Chengdong. On Reflection Corresponding to the Highest Root in Weyl Group; Journal of Tongjiuniversity(Natureal Science) 2006, 34(3): 414-417;
[6] J.Y. Shi. Alcoves corresponding to an affine Weyl group, J. London Math.Soc. 35,1987.
[7] X.G.Zhang. Alcoves of the non Weyl group H_4. to appear Inter natb forum. 2009.
[8] R.W.Carter. Simple Groups of Lie Types, J.Wiley & Sons, London, 1972.
[9] J.E.Humphreys. Introduction to Lie Algebras and Represention Theory [M]. springer-Verlag, Berlin Heidelberg New York, 1972.
[10] Chun Liu, Xi-gou Zhang, Gong-qiang Wang. The Longest Elements and The Highest Roots in the Classical Weyl Groups [J]. Jiangxi Science,26(4):548-551, 2008.
[11] Gong-qiang Wang, Xi-gou Zhang, Chun Liu. The Idempotents of Certain Hecke Algebra [J]. to appear. Journal of JiangXi University(Natureal Sciences Edition), 2009(2).
[12] Thomas W. Hungerford. Algebra. Springer-Verlag, 2003.
[2] R.W.Richardson. Conjugacy classes of involutions in Coxeter groups,Bull. Austral. Math. Soc. 26(1982), 1-15. (68,101).
[3] Meinolf Geck, GOtz Pfeiffer. Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, Clarendon press . oxford,2000.
[4] Gong-qiang Wang, Xi-gou Zhang, Chun Liu. A Special of XiangNan University, 2009(2).
[5] Pu Yanmin, Chen Chengdong. On Reflection Corresponding to the Highest Root in Weyl Group; Journal of Tongjiuniversity(Natureal Science) 2006, 34(3): 414-417;
[6] J.Y. Shi. Alcoves corresponding to an affine Weyl group, J. London Math.Soc. 35,1987.
[7] X.G.Zhang. Alcoves of the non Weyl group H_4. to appear Inter natb forum. 2009.
[8] R.W.Carter. Simple Groups of Lie Types, J.Wiley & Sons, London, 1972.
[9] J.E.Humphreys. Introduction to Lie Algebras and Represention Theory [M]. springer-Verlag, Berlin Heidelberg New York, 1972.
[10] Chun Liu, Xi-gou Zhang, Gong-qiang Wang. The Longest Elements and The Highest Roots in the Classical Weyl Groups [J]. Jiangxi Science,26(4):548-551, 2008.
[11] Gong-qiang Wang, Xi-gou Zhang, Chun Liu. The Idempotents of Certain Hecke Algebra [J]. to appear. Journal of JiangXi University(Natureal Sciences Edition), 2009(2).
[12] Thomas W. Hungerford. Algebra. Springer-Verlag, 2003.