"콕세터 원소(Coxeter element)"의 두 판 사이의 차이
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| − | + | ==개요== | |
| − | + | * 유한 콕세터 군의 특별한 원소들 | |
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| − | * 콕세터 군의 특별한 | ||
* 하나의 conjugacy class를 이룬다 | * 하나의 conjugacy class를 이룬다 | ||
| − | * 원소의 | + | * 원소의 order는 Coxeter number가 된다 |
| + | * quiver의 표현론 등에서 중요한 역할 | ||
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| − | + | ==정의== | |
| − | < | + | * 유한 콕세터 군이 다음과 같이 주어진 경우:<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> |
| + | * 임의의 치환 <math>\pi\in S_{n}</math> 에 대하여, 콕세터 군의 원소 <math>r_{\pi(1)}r_{\pi(2)}\cdots r_{\pi(n)}</math>를 콕세터 원소라 한다 | ||
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| − | + | ==예== | |
| − | * [[ | + | ===대칭군의 콕세터 원소=== |
| + | * [[대칭군 (symmetric group)]] | ||
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| + | ===정이면체군의 콕세터 원소=== | ||
| + | * [[정이면체군(dihedral group)]] | ||
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| − | + | ==역사== | |
| + | * 1951년 콕세터 | ||
| + | * [[수학사 연표]] | ||
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| − | * http://www.ams.org/ | + | ==메모== |
| + | * Michel Deza, Mark Pankov, Zigzag structure of thin chamber complexes, arXiv:1509.03754[math.CO], September 12 2015, http://arxiv.org/abs/1509.03754v2 | ||
| + | * http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=106428&loc=fromrevtext | ||
| + | * Chapuy, Guillaume, and Christian Stump. “Counting Factorizations of Coxeter Elements into Products of Reflections.” arXiv:1211.2789 [math], November 12, 2012. http://arxiv.org/abs/1211.2789. | ||
| + | * Reiner, Victor, Vivien Ripoll, and Christian Stump. “On Non-Conjugate Coxeter Elements in Well-Generated Reflection Groups.” arXiv:1404.5522 [math], April 22, 2014. http://arxiv.org/abs/1404.5522. | ||
* [http://dx.doi.org/10.1016/0021-8693%2889%2990070-7 The spectrum of a Coxeter transformation, affine Coxeter transformations, and the defect map] | * [http://dx.doi.org/10.1016/0021-8693%2889%2990070-7 The spectrum of a Coxeter transformation, affine Coxeter transformations, and the defect map] | ||
* [http://www.math.lsa.umich.edu/%7Ejrs/coxplane.html http://www.math.lsa.umich.edu/~jrs/coxplane.html] | * [http://www.math.lsa.umich.edu/%7Ejrs/coxplane.html http://www.math.lsa.umich.edu/~jrs/coxplane.html] | ||
| − | * | + | * [http://www-igm.univ-mlv.fr/%7Efpsac/FPSAC07/SITE07/Lecture/July3/Nathan%20Reading.pdf http://www-igm.univ-mlv.fr/~fpsac/FPSAC07/SITE07/Lecture/July3/Nathan%20Reading.pdf] |
| − | + | * [http://www.matem.unam.mx/jap/articulos/31.pdf Coxeter transformations and the representation theory of algebras] | |
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| − | + | ==관련된 항목들== | |
| + | * [[유한반사군과 콕세터군(finite reflection groups and Coxeter groups)]] | ||
| + | * [[콕세터 평면]] | ||
| + | * [[바일 벡터]] | ||
| − | + | ==매스매티카 파일 및 계산 리소스== | |
| + | * https://docs.google.com/file/d/0B8XXo8Tve1cxRFNnVmZydjVOQms/edit | ||
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| − | + | ==사전 형태의 자료== | |
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* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
* http://en.wikipedia.org/wiki/Coxeter_plane#Coxeter_plane | * http://en.wikipedia.org/wiki/Coxeter_plane#Coxeter_plane | ||
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| − | + | ==리뷰, 에세이, 강의노트== | |
| − | * http://www.ams.org/ | + | * Bill Casselman, [http://www.ams.org/notices/201108/201108-about-the-cover.pdf The magical Coxeter transformation] Sep 2011 |
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| − | + | ==관련논문== | |
| + | * 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. | ||
| + | * Damianou, Pantelis A., and Charalampos A. Evripidou. “Characteristic and Coxeter Polynomials for Affine Lie Algebras.” arXiv:1409.3956 [math], September 13, 2014. http://arxiv.org/abs/1409.3956. | ||
| + | * Michel, Jean. 2014. “‘Case-Free’ Derivation for Weyl Groups of the Number of Reflection Factorisations of a Coxeter Element.” arXiv:1408.0721 [math], August. http://arxiv.org/abs/1408.0721. | ||
| + | * Ladkani, Sefi. “On the Periodicity of Coxeter Transformations and the Non-Negativity of Their Euler Forms.” Linear Algebra and Its Applications 428, no. 4 (February 1, 2008): 742–53. doi:10.1016/j.laa.2007.08.002. | ||
| + | * Suter, Ruedi. “Coxeter and Dual Coxeter Numbers.” Communications in Algebra 26, no. 1 (1998): 147–53. doi:10.1080/00927879808826122. | ||
| + | * Berman, S, Y. S Lee, and R. V Moody. “The Spectrum of a Coxeter Transformation, Affine Coxeter Transformations, and the Defect Map.” Journal of Algebra 121, no. 2 (March 1989): 339–57. doi:10.1016/0021-8693(89)90070-7. | ||
| + | * Kostant, Bertram. “The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group.” American Journal of Mathematics 81 (1959): 973–1032. | ||
| + | * Coleman, A. J. “The Betti Numbers of the Simple Lie Groups.” Canadian Journal of Mathematics. Journal Canadien de Mathématiques 10 (1958): 349–56. | ||
| + | * The product of the generators of a finite group generated by reflections, HSM Coxeter - Duke Mathematical Journal, 1951 | ||
| + | * Coxeter, H. S. M. “Discrete Groups Generated by Reflections.” Annals of Mathematics, Second Series, 35, no. 3 (July 1, 1934): 588–621. doi:10.2307/1968753. | ||
| + | ** 602p | ||
| + | [[분류:리군과 리대수]] | ||
| − | * | + | ==메타데이터== |
| − | + | ===위키데이터=== | |
| − | * | + | * ID : [https://www.wikidata.org/wiki/Q5179941 Q5179941] |
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'coxeter'}, {'LEMMA': 'element'}] | ||
2021년 2월 17일 (수) 05:02 기준 최신판
개요
- 유한 콕세터 군의 특별한 원소들
- 하나의 conjugacy class를 이룬다
- 원소의 order는 Coxeter number가 된다
- quiver의 표현론 등에서 중요한 역할
정의
- 유한 콕세터 군이 다음과 같이 주어진 경우\[\left\langle r_1,r_2,\ldots,r_n \mid r_1^2=\cdots=r_n^2=(r_ir_j)^{m_{ij}}=1\right\rangle\]
- 임의의 치환 \(\pi\in S_{n}\) 에 대하여, 콕세터 군의 원소 \(r_{\pi(1)}r_{\pi(2)}\cdots r_{\pi(n)}\)를 콕세터 원소라 한다
예
대칭군의 콕세터 원소
정이면체군의 콕세터 원소
역사
- 1951년 콕세터
- 수학사 연표
메모
- Michel Deza, Mark Pankov, Zigzag structure of thin chamber complexes, arXiv:1509.03754[math.CO], September 12 2015, http://arxiv.org/abs/1509.03754v2
- http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=106428&loc=fromrevtext
- Chapuy, Guillaume, and Christian Stump. “Counting Factorizations of Coxeter Elements into Products of Reflections.” arXiv:1211.2789 [math], November 12, 2012. http://arxiv.org/abs/1211.2789.
- Reiner, Victor, Vivien Ripoll, and Christian Stump. “On Non-Conjugate Coxeter Elements in Well-Generated Reflection Groups.” arXiv:1404.5522 [math], April 22, 2014. http://arxiv.org/abs/1404.5522.
- The spectrum of a Coxeter transformation, affine Coxeter transformations, and the defect map
- http://www.math.lsa.umich.edu/~jrs/coxplane.html
- http://www-igm.univ-mlv.fr/~fpsac/FPSAC07/SITE07/Lecture/July3/Nathan%20Reading.pdf
- Coxeter transformations and the representation theory of algebras
관련된 항목들
매스매티카 파일 및 계산 리소스
사전 형태의 자료
리뷰, 에세이, 강의노트
- Bill Casselman, The magical Coxeter transformation Sep 2011
관련논문
- 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.
- Damianou, Pantelis A., and Charalampos A. Evripidou. “Characteristic and Coxeter Polynomials for Affine Lie Algebras.” arXiv:1409.3956 [math], September 13, 2014. http://arxiv.org/abs/1409.3956.
- Michel, Jean. 2014. “‘Case-Free’ Derivation for Weyl Groups of the Number of Reflection Factorisations of a Coxeter Element.” arXiv:1408.0721 [math], August. http://arxiv.org/abs/1408.0721.
- Ladkani, Sefi. “On the Periodicity of Coxeter Transformations and the Non-Negativity of Their Euler Forms.” Linear Algebra and Its Applications 428, no. 4 (February 1, 2008): 742–53. doi:10.1016/j.laa.2007.08.002.
- Suter, Ruedi. “Coxeter and Dual Coxeter Numbers.” Communications in Algebra 26, no. 1 (1998): 147–53. doi:10.1080/00927879808826122.
- Berman, S, Y. S Lee, and R. V Moody. “The Spectrum of a Coxeter Transformation, Affine Coxeter Transformations, and the Defect Map.” Journal of Algebra 121, no. 2 (March 1989): 339–57. doi:10.1016/0021-8693(89)90070-7.
- Kostant, Bertram. “The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group.” American Journal of Mathematics 81 (1959): 973–1032.
- Coleman, A. J. “The Betti Numbers of the Simple Lie Groups.” Canadian Journal of Mathematics. Journal Canadien de Mathématiques 10 (1958): 349–56.
- The product of the generators of a finite group generated by reflections, HSM Coxeter - Duke Mathematical Journal, 1951
- Coxeter, H. S. M. “Discrete Groups Generated by Reflections.” Annals of Mathematics, Second Series, 35, no. 3 (July 1, 1934): 588–621. doi:10.2307/1968753.
- 602p
메타데이터
위키데이터
- ID : Q5179941
Spacy 패턴 목록
- [{'LOWER': 'coxeter'}, {'LEMMA': 'element'}]