"유한반사군과 콕세터 군(finite reflection groups and Coxeter groups)"의 두 판 사이의 차이

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3번째 줄: 3번째 줄:
 
* <math>\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</math>
 
* <math>\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</math>
 
* [[대칭군 (symmetric group)]] 은 콕세터 군의 예이다
 
* [[대칭군 (symmetric group)]] 은 콕세터 군의 예이다
−
** 대칭군 $S_{n+1}$은 $A_n$ 타입의 콕세터 군
+
** 대칭군 <math>S_{n+1}</math>은 <math>A_n</math> 타입의 콕세터 군
 
* [[정이면체군(dihedral group)]]은 콕세터 군의 예이다
 
* [[정이면체군(dihedral group)]]은 콕세터 군의 예이다
−
** 크기가 $2m$인 정이면체 군은 $I_2(m)$ 타입의 콕세터 군
+
** 크기가 <math>2m</math>인 정이면체 군은 <math>I_2(m)</math> 타입의 콕세터 군
 
* 리대수의 이론에 등장하는 바일군(Weyl group) 은 콕세터 군의 예이다
 
* 리대수의 이론에 등장하는 바일군(Weyl group) 은 콕세터 군의 예이다
  
11번째 줄: 11번째 줄:
 
==테이블==
 
==테이블==
 
===분류===
 
===분류===
−
$$
+
:<math>
 
\begin{array}{c|c|c|c|c|c}
 
\begin{array}{c|c|c|c|c|c}
 
   & \text{rank} & \text{degree} & \text{exponent} & \text{order} & \text{Coxeter} \\
 
   & \text{rank} & \text{degree} & \text{exponent} & \text{order} & \text{Coxeter} \\
27번째 줄: 27번째 줄:
 
  I_2(m) & 2 & 2,m & 1,m-1 & 2 m & m
 
  I_2(m) & 2 & 2,m & 1,m-1 & 2 m & m
 
\end{array}
 
\end{array}
−
$$
+
</math>
 
* [[콕세터 군 B3/C3]]
 
* [[콕세터 군 B3/C3]]
 
* [[콕세터 군 H3]]
 
* [[콕세터 군 H3]]
34번째 줄: 34번째 줄:
  
 
===정다면체와 콕세터군===
 
===정다면체와 콕세터군===
−
* $D_4 : 2, 4, 4, 6$
+
* <math>D_4 : 2, 4, 4, 6</math>
−
* $F_4 : 2, 6, 8, 12$
+
* <math>F_4 : 2, 6, 8, 12</math>
−
* $H_4 : 2, 12, 20, 30$
+
* <math>H_4 : 2, 12, 20, 30</math>
  
 
{| style="margin: 1em auto; text-align: center; border-collapse: collapse;"
 
{| style="margin: 1em auto; text-align: center; border-collapse: collapse;"
147번째 줄: 147번째 줄:
  
 
==관련논문==
 
==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
+
* Tomoshige Yukita, On the growth rates of cofinite 3-dimensional Coxeter groups whose dihedral angles are of the form <math>\fracπ{m}</math> for <math>m=2,3,4,5,6</math>, 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.
 
* 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.
 
* 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.
 
* 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.
+
* Bezrukavnikov, Roman, Michael Finkelberg, and Ivan Mirković. “Equivariant (<math>K</math>-)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.
 
* 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.
 
* 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.
 
* 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
+
* H. S. M. Coxeter, The complete enumeration of finite groups of the form <math>R^2_i = (R_iR_j)^{k_{ij}} = 1</math>, 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.
 
* Coxeter, H. S. M. ‘Discrete Groups Generated by Reflections’. Annals of Mathematics. Second Series 35, no. 3 (1934): 588–621. doi:10.2307/1968753.
  

2020년 11월 16일 (월) 05:06 판

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


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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.
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