"Bruhat decomposition"의 두 판 사이의 차이

2021년 2월 17일 (수) 01:38 기준 최신판

introduction

Given a Lie group \(G\) over \(\mathbb{C}\) and a Borel subgroup \(B\), there is famous Bruhat decomposition of the flag variety \(G/B\)
\(G\) : 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 \(B\)

we say \((B,N)\) form a \(BN\)-pair of \(G\), or that \((G,B,N,S)\) is a Tits system
we call \(B\) the Borel subgroup of \(G\), and \(W=N/B\cap N\) the Weyl group associated with the Tits system
the rank of the Tits system is defined to be \(|S|\)

why do we care?

\((B, N)\) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs.
Roughly speaking, it shows that all such groups are similar to the general linear group over a field
BN-pairs can be used to prove that most groups of Lie type are simple

Bruhat decomposition theorem

thm

Let \(G\) be a group with a \(BN\)-pair. Then \[ G=BWB \] or, \[ G=\cup_{w\in W}BwB \] in which the union is disjoint, where \(BwB\) is taken to mean \(B\dot{w}B\) for any \(\dot{w}\in N\) with \(\dot{w}T=w\)

memo

http://qchu.wordpress.com/2010/07/11/chevalley-bruhat-order/

related items

Cartan decomposition of general linear groups

computational resource

https://docs.google.com/file/d/0B8XXo8Tve1cxZzFwSzhRYnRHalE/edit

encyclopedia

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.

question and answers(Math Overflow)

메타데이터

위키데이터

ID : Q4978699

Spacy 패턴 목록

[{'LOWER': 'bruhat'}, {'LEMMA': 'decomposition'}]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* $G$ : connected reductive algebraic group over an algebraically closed field
+* Given a Lie group <math>G</math> over <math>\mathbb{C}</math> and a Borel subgroup <math>B</math>, there is famous Bruhat decomposition of the flag variety <math>G/B</math>
-* 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 $G$
+* <math>G</math> : connected reductive algebraic group over an algebraically closed field
+* By allowing one to reduce many questions about <math>G</math> to questions about the Weyl group <math>W</math>, Bruhat decomposition is indispensable for the understanding of both the structure and representations of <math>G</math>
 * 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==
+==example : general linear group==
-* $G=GL_{n}$
+* <math>G=GL_{n}</math>
-* $B$ : upper triangular matrices in $G$
+* <math>B</math> : upper triangular matrices in <math>G</math>
-* $B_{-}$ : lower triangular matrices in $G$
+* <math>B_{-}</math> : lower triangular matrices in <math>G</math>
-* $W=S_{n}$ we can think of it as a subgroup of $G$
+* <math>W=S_{n}</math> we can think of it as a subgroup of <math>G</math>
-* Double cosets <math>BwB</math> and <math>B_{-}wB_{-}</math> are called Bruhat cells.
+* Double cosets <math>BwB</math> and <math>B_{-}wB_{-}</math> are called Bruhat cells.
+==(B, N) pair==
+* A <math>(B, N)</math> pair is a pair of subgroups <math>B</math> and <math>N</math> of a group <math>G</math> such that the following axioms hold:
+# <math>G</math> is generated by <math>B</math> and <math>N</math>
+# The intersection, <math>T</math>, of <math>B</math> and <math>N</math> is a normal subgroup of N
+# The group <math>W = N/T</math> is generated by a set <math>S</math> of elements <math>w_i</math> of order 2, for <math>i</math> in some non-empty set <math>I</math>
+# If <math>w_i</math> is an element of <math>S</math> and <math>w</math> is any element of <math>W</math>, then <math>w_iBw</math> is contained in the union of <math>Bw_iwB</math> and <math>BwB</math>
+# No generator <math>w_i</math> normalizes <math>B</math>
+* we say <math>(B,N)</math> form a <math>BN</math>-pair of <math>G</math>, or that <math>(G,B,N,S)</math> is a Tits system
+* we call <math>B</math> the Borel subgroup of <math>G</math>, and <math>W=N/B\cap N</math> the Weyl group associated with the Tits system
+* the rank of the Tits system is defined to be <math>|S|</math>
+===why do we care?===
+* <math>(B, N)</math> pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs.
+* Roughly speaking, it shows that all such groups are similar to the general linear group over a field
+* BN-pairs can be used to prove that most groups of Lie type are simple
+==Bruhat decomposition theorem==
+;thm
+Let <math>G</math> be a group with a <math>BN</math>-pair. Then
+:<math>
+G=BWB
+</math>
+or,
+:<math>
+G=\cup_{w\in W}BwB
+</math>
+in which the union is disjoint, where <math>BwB</math> is taken to mean <math>B\dot{w}B</math> for any <math>\dot{w}\in N</math> with <math>\dot{w}T=w</math>
@@ 22번째 줄: / 51번째 줄: @@
+==related items==
+* [[Cartan decomposition of general linear groups]]
 ==computational resource==
 * https://docs.google.com/file/d/0B8XXo8Tve1cxZzFwSzhRYnRHalE/edit
 ==encyclopedia==
+* http://en.wikipedia.org/wiki/(B,_N)_pair
 * http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group
 * http://eom.springer.de/b/b017690.htm
@@ 41번째 줄: / 72번째 줄: @@
 * [http://www.ryancreich.info/bruhat_row-reduction.pdf Bruhat decomposition via row reduction]
 ==articles==
 * '''[C1]''' Chevalley, C. 1955. “Sur Certains Groupes Simples.” The Tohoku Mathematical Journal. Second Series 7: 14–66.
@@ 51번째 줄: / 82번째 줄: @@
 * http://mathoverflow.net/questions/15438/a-slick-proof-of-the-bruhat-decomposition-for-gl-nk
 * http://mathoverflow.net/questions/28569/is-there-a-morse-theory-proof-of-the-bruhat-decomposition
+* http://mathoverflow.net/questions/168033/coxeter-groups-parabolic-subgroups/168035#168035
+* http://mathoverflow.net/questions/188920/closure-relations-between-bruhat-cells-on-the-flag-variety/190961#190961
 [[분류:개인노트]]
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 [[분류:math and physics]]
 [[분류:math]]
+[[분류:migrate]]
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q4978699 Q4978699]
+===Spacy 패턴 목록===
+* [{'LOWER': 'bruhat'}, {'LEMMA': 'decomposition'}]