유한반사군과 콕세터 군(finite reflection groups and Coxeter groups)

개요

• \(\left\langle r_1,r_2,\ldots,r_n \mid r_1^2=\cdots=r_n^2=(r_ir_j)^{m_{ij}}=1\right\rangle\)
• 대칭군 (symmetric group) 은 콕세터 군의 예이다
  • 대칭군 $S_{n+1}$은 $A_n$ 타입의 콕세터 군
• 정이면체군(dihedral group)은 콕세터 군의 예이다
  • 크기가 $2m$인 정이면체 군은 $I_2(m)$ 타입의 콕세터 군
• 리대수의 이론에 등장하는 바일군(Weyl group) 은 콕세터 군의 예이다

테이블

분류

$$ \begin{array}{c|c|c|c|c|c} & \text{rank} & \text{degree} & \text{exponent} & \text{order} & \text{Coxeter} \\ \hline A_n & n & 2,3,\cdots, n+1 & 1,2,\cdots, n& (n+1)! & n+1 \\ B_n/C_n & n & 2,4,6,\cdots,2n & 1,3,5,\cdots,2n-1 & 2^n n! & 2 n \\ D_n & n & 2,4,6,\cdots 2n-2, n & 1,3,5,\cdots,2n-3, n-1 & 2^{n-1} n! & 2 n-2 \\ E_6 & 6 & 2,5,6,8,9,12 & 1,4,5,7,8,11 & 51840 & 12 \\ E_7 & 7 & 2,6,8,10,12,14,18 & 1,5,7,9,11,13,17 & 2903040 & 18 \\ E_8 & 8 & 2,8,12,14,18,20,24,30 & 1,7,11,13,17,19,23,29 & 696729600 & 30 \\ F_4 & 4 & 2,6,8,12 & 1,5,7,11 & 1152 & 12 \\ G_2 & 2 & 2,6 & 1,5 & 12 & 6 \\ H_3 & 3 & 2,6,10 & 1,5,9 & 120 & 10 \\ H_4 & 4 & 2,12,20,30 & 1,11,19,29 & 14400 & 30 \\ I_2(m) & 2 & 2,m & 1,m-1 & 2 m & m \end{array} $$

• 콕세터 군 B3/C3
• 콕세터 군 H3
• 콕세터 군 H4
• 콕세터 군의 차수와 지수 (degrees and exponents)

정다면체와 콕세터군

• $D_4 : 2, 4, 4, 6$
• $F_4 : 2, 6, 8, 12$
• $H_4 : 2, 12, 20, 30$

다면체	점 V	선 E	면 F	V-E+F
정사면체	4	6	4	4-6+4=2
정육면체	8	12	6	8-12+6=2
정팔면체	6	12	8	6-12+8=2
정십이면체	20	30	12	20-30+12=2
정이십면체	12	30	20	12-30+20=2

역사

• Élie Cartan
• 1934 콕세터
• 수학사 연표

메모

• 정다면체와의 숨바꼭질, 피타고라스의 창, 2009-2-11
• http://mathoverflow.net/questions/188980/what-are-the-outer-automorphisms-of-a-coxeter-group
• Morin-Duchesne, Alexi, Jorgen Rasmussen, and Philippe Ruelle. “Dimer Representations of the Temperley-Lieb Algebra.” arXiv:1409.3416 [cond-Mat, Physics:hep-Th, Physics:math-Ph], September 11, 2014. http://arxiv.org/abs/1409.3416.
• 강의록 http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html
• 비디오 강의 http://vod.mathnet.or.kr/sub4_1.php?key_s_title=Coxeter+Groups+and+Reflection+Symmetry+Ten+Lectures+by+Jon+McCammond&key_year=x

관련된 항목들

• 콕세터 원소(Coxeter element)
• 반사 변환
• 정다면체
• 5차방정식과 정이십면체
• 몰리엔 정리 (Molien's theorem)
• Limit roots of infinite Coxeter groups

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxcjdIZUFISk0wajA/edit

사전 형태의 자료

• http://en.wikipedia.org/wiki/Reflection_group
• http://en.wikipedia.org/wiki/Coxeter_group
• http://en.wikipedia.org/wiki/Chevalley–Shephard–Todd_theorem
• http://www.encyclopediaofmath.org/index.php/Coxeter_group

리뷰, 에세이, 강의노트

• Belolipetsky, Mikhail. “Arithmetic Hyperbolic Reflection Groups.” arXiv:1506.03111 [math], June 9, 2015. http://arxiv.org/abs/1506.03111.
• Rouquier, Weyl groups, affine Weyl groups and reflection groups
• Arjeh M. Cohen, Coxeter groups http://www.win.tue.nl/~jpanhuis/coxeter/notes/notes.pdf
• Heckman, Gert. "Coxeter Groups." Lecture Notes, Fall (2013). http://www.math.ru.nl/~heckman/CoxeterGroups.pdf
• Daniel Allcock 'The finite reflection groups'
  • Daniel Allcock expository articles
• Roe Goodman Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes, The American Mathematical Monthly, Vol. 111, No. 4 (Apr., 2004), pp. 281-298
• Bourbaki, Nicolas. ‘Groups Generated by Reflections; Root Systems’. In Elements of the History of Mathematics, 269–74. Springer Berlin Heidelberg, 1994. http://link.springer.com.ezproxy.library.uq.edu.au/chapter/10.1007/978-3-642-61693-8_26.
• Logothetti, Dave, and H. S. M. Coxeter. ‘An Interview with H. S. M. Coxeter, the King of Geometry’. The Two-Year College Mathematics Journal 11, no. 1 (1 January 1980): 2–19. doi:10.2307/3026700.

관련논문

• Bezrukavnikov, Roman, Michael Finkelberg, and Ivan Mirković. “Equivariant ($K$-)homology of Affine Grassmannian and Toda Lattice.” arXiv:math/0306413, June 29, 2003. http://arxiv.org/abs/math/0306413.
• Kato, Mitsuo, and Jiro Sekiguchi. “Regular Polyhedral Groups and Reflection Groups of Rank Four.” European Journal of Combinatorics, Arithmetique et Combinatoire, 25, no. 4 (May 2004): 565–77. doi:10.1016/j.ejc.2003.09.013.
• Steinberg, Robert. “Finite Reflection Groups.” Transactions of the American Mathematical Society 91 (1959): 493–504.
• Chevalley, Claude. “Invariants of Finite Groups Generated by Reflections.” American Journal of Mathematics 77 (1955): 778–82.
• H. S. M. Coxeter, The complete enumeration of finite groups of the form $R^2_i = (R_iR_j)^{k_{ij}} = 1$, J. London Math. Soc. 10 (1935), 21–25
• Coxeter, H. S. M. ‘Discrete Groups Generated by Reflections’. Annals of Mathematics. Second Series 35, no. 3 (1934): 588–621. doi:10.2307/1968753.