"Bruhat decomposition"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| − | == | + | ==example : general linear group== |
* $G=GL_{n}$ | * $G=GL_{n}$ | ||
* $B$ : upper triangular matrices in $G$ | * $B$ : upper triangular matrices in $G$ | ||
| 16번째 줄: | 16번째 줄: | ||
| + | ==(B, N) pair== | ||
| + | * A $(B, N)$ pair is a pair of subgroups $B$ and $N$ of a group $G$ such that the following axioms hold: | ||
| + | # $G$ is generated by $B$ and $N$ | ||
| + | # The intersection, $T$, of $B$ and $N$ is a normal subgroup of N | ||
| + | # The group $W = N/T$ is generated by a set $S$ of elements $w_i$ of order 2, for $i$ in some non-empty set $I$ | ||
| + | # If $w_i$ is an element of $S$ and $w$ is any element of $W$, then $w_iBw$ is contained in the union of $Bw_iwB$ and $BwB$ | ||
| + | # No generator $w_i$ normalizes $B$ | ||
| + | |||
| + | |||
| + | |||
| + | ==Bruhat decomposition theorem== | ||
;thm | ;thm | ||
| + | Let $G$ be a group with a $BN$-pair. Then | ||
$$ | $$ | ||
| − | G= | + | G=\union_{w\in W}BwB |
$$ | $$ | ||
| + | in which the union is disjoint | ||
| + | |||
2014년 3월 27일 (목) 20:04 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
example : general linear group
- $G=GL_{n}$
- $B$ : 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.
(B, N) pair
- A $(B, N)$ pair is a pair of subgroups $B$ and $N$ of a group $G$ such that the following axioms hold:
- $G$ is generated by $B$ and $N$
- The intersection, $T$, of $B$ and $N$ is a normal subgroup of N
- The group $W = N/T$ is generated by a set $S$ of elements $w_i$ of order 2, for $i$ in some non-empty set $I$
- If $w_i$ is an element of $S$ and $w$ is any element of $W$, then $w_iBw$ is contained in the union of $Bw_iwB$ and $BwB$
- No generator $w_i$ normalizes $B$
Bruhat decomposition theorem
- thm
Let $G$ be a group with a $BN$-pair. Then $$ G=\union_{w\in W}BwB $$ in which the union is disjoint
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.