"Bruhat decomposition"의 두 판 사이의 차이
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* $G$ : connected reductive algebraic group over an algebraically closed field | * $G$ : connected reductive algebraic group over an algebraically closed field | ||
* By allowing one to reduce many questions about $G$ to questions about the Weyl group $W$, Bruhat decomposition is indispensable for the understanding of both the structure and representations of $G$ | * By allowing one to reduce many questions about $G$ to questions about the Weyl group $W$, Bruhat decomposition is indispensable for the understanding of both the structure and representations of $G$ | ||
| − | * The order of a Chevalley group over a finite field was computed in [C1] (using Bruhat decomposition) in terms of the exponents of the Weyl group | + | * The order of a Chevalley group over a finite field was computed in '''[C1]''' (using Bruhat decomposition) in terms of the exponents of the Weyl group |
* Bruhat order | * Bruhat order | ||
* Weyl group action | * Weyl group action | ||
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==articles== | ==articles== | ||
| − | * Chevalley, C. 1955. “Sur Certains Groupes Simples.” The Tohoku Mathematical Journal. Second Series 7: 14–66. | + | * '''[C1]''' Chevalley, C. 1955. “Sur Certains Groupes Simples.” The Tohoku Mathematical Journal. Second Series 7: 14–66. |
* Bruhat, Fran\ccois. 1956. “Sur Les Représentations Induites Des Groupes de Lie.” Bulletin de La Société Mathématique de France 84: 97–205. | * Bruhat, Fran\ccois. 1956. “Sur Les Représentations Induites Des Groupes de Lie.” Bulletin de La Société Mathématique de France 84: 97–205. | ||
2013년 12월 7일 (토) 07:51 판
introduction
- $G$ : connected reductive algebraic group over an algebraically closed field
- By allowing one to reduce many questions about $G$ to questions about the Weyl group $W$, Bruhat decomposition is indispensable for the understanding of both the structure and representations of $G$
- The order of a Chevalley group over a finite field was computed in [C1] (using Bruhat decomposition) in terms of the exponents of the Weyl group
- Bruhat order
- Weyl group action
Bruhat cell
- $G=GL_{n}$
- $B$ : upper triangular matrices in $G$
- $B_{-}$ : lower triangular matrices in $G$
- $W=S_{n}$ we can think of it as a subgroup of $G$
- Double cosets \(BwB\) and \(B_{-}wB_{-}\) are called Bruhat cells.
memo
computational resource
encyclopedia
- http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group
- http://eom.springer.de/b/b017690.htm
expositions
- Lusztig, G. 2010. “Bruhat Decomposition and Applications.” arXiv:1006.5004 [math] (June 25). http://arxiv.org/abs/1006.5004.
- http://math.ucr.edu/home/baez/week186.html
- Bruhat decomposition via row reduction
articles
- [C1] Chevalley, C. 1955. “Sur Certains Groupes Simples.” The Tohoku Mathematical Journal. Second Series 7: 14–66.
- Bruhat, Fran\ccois. 1956. “Sur Les Représentations Induites Des Groupes de Lie.” Bulletin de La Société Mathématique de France 84: 97–205.