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

수학노트
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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
  
  
 
==테이블==
 
==테이블==
−
===불변량===
+
===분류===
−
$$
+
:<math>
−
\begin{array}{c|ccccc}
+
\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} \\
 
\hline
 
\hline
 
  A_n & n & 2,3,\cdots, n+1 & 1,2,\cdots, n&  (n+1)! & n+1 \\
 
  A_n & n & 2,3,\cdots, n+1 & 1,2,\cdots, n&  (n+1)! & n+1 \\
−
  B_n & n & 2,4,6,\cdots,2n & 1,3,5,\cdots,2n-1 & 2^n n! & 2 n \\
+
  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 \\
 
  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_6 & 6 & 2,5,6,8,9,12 & 1,4,5,7,8,11 & 51840 & 12 \\
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 H3]]
 
* [[콕세터 군 H3]]
−
 
+
* [[콕세터 군 H4]]
 +
* [[콕세터 군의 차수와 지수 (degrees and exponents)]]
  
 
===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;"
103번째 줄: 105번째 줄:
  
 
==메모==
 
==메모==
−
 
+
* [http://bomber0.byus.net/index.php/2009/02/11/1009 정다면체와의 숨바꼭질], 피타고라스의 창, 2009-2-11
−
* Arjeh M. Cohen Coxeter groups [http://www.win.tue.nl/%7Ejpanhuis/coxeter/notes/notes.pdf http://www.win.tue.nl/~jpanhuis/coxeter/notes/notes.pdf]
+
* http://mathoverflow.net/questions/188980/what-are-the-outer-automorphisms-of-a-coxeter-group
 +
* Lange, Christian, and Marina A. Mikhailova. “Classification of Finite Groups Generated by Reflections and Rotations.” arXiv:1509.06922 [math], September 23, 2015. http://arxiv.org/abs/1509.06922.
 +
* 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
 
* 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
 
* 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==
 
==관련된 항목들==
 
* [[콕세터 원소(Coxeter element)]]
 
* [[콕세터 원소(Coxeter element)]]
 +
* [[반사 변환]]
 
* [[정다면체]]
 
* [[정다면체]]
 
* [[5차방정식과 정이십면체]]
 
* [[5차방정식과 정이십면체]]
 
* [[몰리엔 정리 (Molien's theorem)]]
 
* [[몰리엔 정리 (Molien's theorem)]]
−
 
+
* [[Limit roots of infinite Coxeter groups]]
 +
* [[쌍곡 콕세터 군]]
 +
* [[콕세터 군에서의 축약 표현]]
 +
* [[콕세터 군의 표현론]]
 +
* [[콕세터 군에서의 알고리즘]]
  
 
==매스매티카 파일 및 계산 리소스==
 
==매스매티카 파일 및 계산 리소스==
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxcjdIZUFISk0wajA/edit
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxcjdIZUFISk0wajA/edit
−
 
  
 
==사전 형태의 자료==
 
==사전 형태의 자료==
127번째 줄: 134번째 줄:
 
   
 
   
  
−
==관련논문==
+
==리뷰, 에세이, 강의노트==
−
 
+
* Belolipetsky, Mikhail. “Arithmetic Hyperbolic Reflection Groups.” arXiv:1506.03111 [math], June 9, 2015. http://arxiv.org/abs/1506.03111.
−
* Mitsuo Kato and Jiro Sekiguchi [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WDY-4B0WHXW-1&_user=4420&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059607&_version=1&_urlVersion=0&_userid=4420&md5=188db4d982dbbcd13fb099e37f43bc91 Regular polyhedral groups and reflection groups of rank four]
+
* Geck, Meinolf. “PyCox: Computing with (finite) Coxeter Groups and Iwahori-Hecke Algebras.” LMS Journal of Computation and Mathematics 15 (November 2012): 231–56. doi:10.1112/S1461157012001064.
−
* Roe Goodman [http://www.jstor.org/stable/i387719 Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes], <cite>The American Mathematical Monthly</cite>, Vol. 111, No. 4 (Apr., 2004), pp. 281-298
+
* Rouquier, [http://people.maths.ox.ac.uk/rouquier/papers/weyl.pdf Weyl groups, affine Weyl groups and reflection groups]
 +
* Arjeh M. Cohen, [http://www.win.tue.nl/~amc/pub/CoxNotes.pdf Coxeter groups]
 +
* Heckman, Gert. "Coxeter Groups." Lecture Notes, Fall (2013). http://www.math.ru.nl/~heckman/CoxeterGroups.pdf
 
* Daniel Allcock '[http://www.ma.utexas.edu/users/allcock/expos/reflec_classification.pdf The finite reflection groups]'
 
* Daniel Allcock '[http://www.ma.utexas.edu/users/allcock/expos/reflec_classification.pdf The finite reflection groups]'
 
** [http://www.ma.utexas.edu/users/allcock/ Daniel Allcock expository articles]
 
** [http://www.ma.utexas.edu/users/allcock/ Daniel Allcock expository articles]
 +
* Roe Goodman [http://www.jstor.org/stable/i387719 Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes], <cite>The American Mathematical Monthly</cite>, 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==
 +
* 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.
 +
* 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 (<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.
 
* 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.
  
  
−
==블로그==
 
−
* [http://bomber0.byus.net/index.php/2009/02/11/1009 정다면체와의 숨바꼭질]
 
−
** 피타고라스의 창, 2009-2-11
 
 
[[분류:리군과 리대수]]
 
[[분류:리군과 리대수]]
 
[[분류:테셀레이션]]
 
[[분류:테셀레이션]]
 +
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q7307244 Q7307244]
 +
===Spacy 패턴 목록===
 +
* [{'LOWER': 'reflection'}, {'LEMMA': 'group'}]
 +
* [{'LOWER': 'reflection'}, {'LEMMA': 'group'}]

2021년 2월 17일 (수) 05:56 기준 최신판

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


역사


메모

관련된 항목들

매스매티카 파일 및 계산 리소스

사전 형태의 자료


리뷰, 에세이, 강의노트

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.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'reflection'}, {'LEMMA': 'group'}]
  • [{'LOWER': 'reflection'}, {'LEMMA': 'group'}]
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