Bruhat decomposition - 편집 역사

2021년 2월 17일 (수) 09:38에 Pythagoras0님의 편집

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2020년 12월 28일 (월) 12:17에 Pythagoras0님의 편집

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Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T02:53:55Z

메타데이터: 새 문단

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2020년 11월 13일 (금) 15:06에 imported>Pythagoras0님의 편집

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imported>Pythagoras0: /* memo */

2020-02-19T07:28:06Z

memo

imported>Pythagoras0: /* question and answers(Math Overflow) */

2014-12-29T13:43:17Z

question and answers(Math Overflow)

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2014년 5월 25일 (일) 15:04에 imported>Pythagoras0님의 편집

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2014년 3월 28일 (금) 04:16에 imported>Pythagoras0님의 편집

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@@ 91번째 줄: / 91번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q4978699 Q4978699]

@@ 5번째 줄: / 5번째 줄: @@
 * 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
@@ 14번째 줄: / 14번째 줄: @@
 * <math>B_{-}</math> : lower triangular matrices in <math>G</math>
 * <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.
@@ 56번째 줄: / 56번째 줄: @@
 ==computational resource==
 * https://docs.google.com/file/d/0B8XXo8Tve1cxZzFwSzhRYnRHalE/edit
-−
@@ 72번째 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.

← 이전 판		2020년 12월 28일 (월) 02:53 판
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	[[분류:math]]		[[분류:math]]
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q4978699 Q4978699]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$, there is famous Bruhat decomposition of the flag variety $G/B$
+* 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>
-* $G$ : connected reductive algebraic group over an algebraically closed field
+* <math>G</math> : 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 <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
@@ 10번째 줄: / 10번째 줄: @@
 ==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.
 ==(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:
+* 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:
-# $G$ is generated by $B$ and $N$
+# <math>G</math> is generated by <math>B</math> and <math>N</math>
-# The intersection, $T$, of $B$ and $N$ is a normal subgroup of N
+# The intersection, <math>T</math>, of <math>B</math> and <math>N</math> 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 $I$
+# 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 $w_i$ is an element of $S$ and $w$ is any element of $W$, then $w_iBw$ is contained in the union of $Bw_iwB$ and $BwB$
+# 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 $w_i$ normalizes $B$
+# No generator <math>w_i</math> normalizes <math>B</math>
-* we say $(B,N)$ form a $BN$-pair of $G$, or that $(G,B,N,S)$ is a Tits system
+* 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 $B$ the Borel subgroup of $G$, and $W=N/B\cap N$ the Weyl group associated with the 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 $|S|$
+* the rank of the Tits system is defined to be <math>|S|</math>
 ===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.
+* <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
@@ 35번째 줄: / 35번째 줄: @@
 ==Bruhat decomposition theorem==
 ;thm
-Let $G$ be a group with a $BN$-pair. Then
+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 $BwB$ is taken to mean $B\dot{w}B$ for any $\dot{w}\in N$ with $\dot{w}T=w$
+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>

← 이전 판		2020년 11월 13일 (금) 15:06 판
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	[[분류:math]]		[[분류:math]]
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← 이전 판		2020년 2월 19일 (수) 07:28 판
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		+	==related items==
		+	* [[Cartan decomposition of general linear groups]]

	==computational resource==		==computational resource==

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 * 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
 [[분류:개인노트]]

← 이전 판		2014년 3월 28일 (금) 04:16 판
26번째 줄:		26번째 줄:
	* we call $B$ the Borel subgroup of $G$, and $W=N/B\cap N$ the Weyl group associated with the 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 $\|S\|$		* 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