"유한반사군과 콕세터 군(finite reflection groups and Coxeter groups)"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) (→관련논문) |
Pythagoras0 (토론 | 기여) |
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(같은 사용자의 중간 판 26개는 보이지 않습니다) | |||
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 | + | ** 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 | + | ** 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| | + | \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 | ||
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]] | ||
* [[콕세터 군 H4]] | * [[콕세터 군 H4]] | ||
* [[콕세터 군의 차수와 지수 (degrees and exponents)]] | * [[콕세터 군의 차수와 지수 (degrees and exponents)]] | ||
− | |||
===정다면체와 콕세터군=== | ===정다면체와 콕세터군=== | ||
− | * | + | * <math>D_4 : 2, 4, 4, 6</math> |
− | * | + | * <math>F_4 : 2, 6, 8, 12</math> |
− | * | + | * <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;" | ||
106번째 줄: | 105번째 줄: | ||
==메모== | ==메모== | ||
+ | * [http://bomber0.byus.net/index.php/2009/02/11/1009 정다면체와의 숨바꼭질], 피타고라스의 창, 2009-2-11 | ||
+ | * 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. | * 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 | ||
117번째 줄: | 118번째 줄: | ||
* [[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== | ||
130번째 줄: | 135번째 줄: | ||
==리뷰, 에세이, 강의노트== | ==리뷰, 에세이, 강의노트== | ||
+ | * Belolipetsky, Mikhail. “Arithmetic Hyperbolic Reflection Groups.” arXiv:1506.03111 [math], June 9, 2015. http://arxiv.org/abs/1506.03111. | ||
+ | * 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. | ||
+ | * 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 | * 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== | ==관련논문== | ||
− | * Bezrukavnikov, Roman, Michael Finkelberg, and Ivan Mirković. “Equivariant ( | + | * 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. | * 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 | + | * 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. | ||
+ | |||
− | |||
− | |||
− | |||
[[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
- 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
- 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
- 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
- 쌍곡 콕세터 군
- 콕세터 군에서의 축약 표현
- 콕세터 군의 표현론
- 콕세터 군에서의 알고리즘
매스매티카 파일 및 계산 리소스
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.
- 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.
- Rouquier, Weyl groups, affine Weyl groups and reflection groups
- Arjeh M. Cohen, Coxeter groups
- Heckman, Gert. "Coxeter Groups." Lecture Notes, Fall (2013). http://www.math.ru.nl/~heckman/CoxeterGroups.pdf
- Daniel Allcock 'The finite reflection groups'
- 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
- 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
- ID : Q7307244
Spacy 패턴 목록
- [{'LOWER': 'reflection'}, {'LEMMA': 'group'}]
- [{'LOWER': 'reflection'}, {'LEMMA': 'group'}]