콕세터 원소(Coxeter element)

개요

• 유한 콕세터 군의 특별한 원소들
• 하나의 conjugacy class를 이룬다
• 원소의 order는 Coxeter number가 된다
• quiver의 표현론 등에서 중요한 역할

 

정의

• 유한 콕세터 군이 다음과 같이 주어진 경우\[\left\langle r_1,r_2,\ldots,r_n \mid r_1^2=\cdots=r_n^2=(r_ir_j)^{m_{ij}}=1\right\rangle\]
• 임의의 치환 \(\pi\in S_{n}\) 에 대하여, 콕세터 군의 원소 \(r_{\pi(1)}r_{\pi(2)}\cdots r_{\pi(n)}\)를 콕세터 원소라 한다

 

예

대칭군의 콕세터 원소

• 대칭군 (symmetric group)

 

정이면체군의 콕세터 원소

• 정이면체군(dihedral group)

 

역사

• 1951년 콕세터
• 수학사 연표

 

메모

• http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=106428&loc=fromrevtext
• Chapuy, Guillaume, and Christian Stump. “Counting Factorizations of Coxeter Elements into Products of Reflections.” arXiv:1211.2789 [math], November 12, 2012. http://arxiv.org/abs/1211.2789.
• Reiner, Victor, Vivien Ripoll, and Christian Stump. “On Non-Conjugate Coxeter Elements in Well-Generated Reflection Groups.” arXiv:1404.5522 [math], April 22, 2014. http://arxiv.org/abs/1404.5522.
• The spectrum of a Coxeter transformation, affine Coxeter transformations, and the defect map
• http://www.math.lsa.umich.edu/~jrs/coxplane.html
• http://www-igm.univ-mlv.fr/~fpsac/FPSAC07/SITE07/Lecture/July3/Nathan%20Reading.pdf
• Coxeter transformations and the representation theory of algebras

관련된 항목들

• 유한반사군과 콕세터군(finite reflection groups and Coxeter groups)
• 콕세터 평면

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxRFNnVmZydjVOQms/edit

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Coxeter_plane#Coxeter_plane

리뷰, 에세이, 강의노트

• Bill Casselman, The magical Coxeter transformation Sep 2011

관련논문

• Damianou, Pantelis A., and Charalampos A. Evripidou. “Characteristic and Coxeter Polynomials for Affine Lie Algebras.” arXiv:1409.3956 [math], September 13, 2014. http://arxiv.org/abs/1409.3956.
• Michel, Jean. 2014. “‘Case-Free’ Derivation for Weyl Groups of the Number of Reflection Factorisations of a Coxeter Element.” arXiv:1408.0721 [math], August. http://arxiv.org/abs/1408.0721.
• Ladkani, Sefi. “On the Periodicity of Coxeter Transformations and the Non-Negativity of Their Euler Forms.” Linear Algebra and Its Applications 428, no. 4 (February 1, 2008): 742–53. doi:10.1016/j.laa.2007.08.002.
• Suter, Ruedi. “Coxeter and Dual Coxeter Numbers.” Communications in Algebra 26, no. 1 (1998): 147–53. doi:10.1080/00927879808826122.
• Berman, S, Y. S Lee, and R. V Moody. “The Spectrum of a Coxeter Transformation, Affine Coxeter Transformations, and the Defect Map.” Journal of Algebra 121, no. 2 (March 1989): 339–57. doi:10.1016/0021-8693(89)90070-7.
• Kostant, Bertram. “The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group.” American Journal of Mathematics 81 (1959): 973–1032.
• Coleman, A. J. “The Betti Numbers of the Simple Lie Groups.” Canadian Journal of Mathematics. Journal Canadien de Mathématiques 10 (1958): 349–56.
• The product of the generators of a finite group generated by reflections, HSM Coxeter - Duke Mathematical Journal, 1951
• Coxeter, H. S. M. “Discrete Groups Generated by Reflections.” Annals of Mathematics, Second Series, 35, no. 3 (July 1, 1934): 588–621. doi:10.2307/1968753.
  • 602p