# 유한반사군과 콕세터 군(finite reflection groups and Coxeter groups)

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## 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

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

### 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

## 관련논문

• Tomoshige Yukita, On the growth rates of cofinite 3-dimensional Coxeter groups whose dihedral angles are of the form $\fracπ{m}$ for $m=2,3,4,5,6$, http://arxiv.org/abs/1603.04592v1
• Kamgarpour, Masoud. “Stabilisers of Eigenvectors of Finite Reflection Groups.” arXiv:1512.01591 [math], December 4, 2015. http://arxiv.org/abs/1512.01591.
• Labbé, Jean-Philippe, and Sébastien Labbé. “A Perron Theorem for Matrices with Negative Entries and Applications to Coxeter Groups.” arXiv:1511.04975 [math], November 16, 2015. http://arxiv.org/abs/1511.04975.
• Deza, Michel, and Mark Pankov. “Zigzag Structure of Thin Chamber Complexes.” arXiv:1509.03754 [math], September 12, 2015. http://arxiv.org/abs/1509.03754.
• 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.