"Bruhat decomposition"의 두 판 사이의 차이

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==introduction==
 
==introduction==
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* double Bruhat cells
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* 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>
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* <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
 
* Bruhat order
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* Weyl group action 
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* Weyl group action
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* The decomposition of G into strata G^{u,v} is 'good with respect to total positivity.
 
  
  
  
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==Bruhat cell==
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==example : general linear group==
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* $G=GL_{n}$
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* <math>G=GL_{n}</math>
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* $B : upper triangular matrices in $G$
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* <math>B</math> : upper triangular matrices in <math>G</math>
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* $B_{_}$ : lower triangular matrices in $G$
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* <math>B_{-}</math> : lower triangular matrices in <math>G</math>
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* $W=S_{n}$ we can think of it as a subgroup of $G$
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* <math>W=S_{n}</math> we can think of it as a subgroup of <math>G</math>
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* Double cosets <math>BwB</math> and <math>B_{-}wB_{-}</math> are called Bruhat cells.
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* Double cosets <math>BwB</math> and <math>B_{-}wB_{-}</math> are called Bruhat cells.
  
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==double Bruhat cell (DBC)==
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==(B, N) pair==
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* 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:
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# <math>G</math> is generated by <math>B</math> and <math>N</math>
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# The intersection, <math>T</math>, of <math>B</math> and <math>N</math> is a normal subgroup of N
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# 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>
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# 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>
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# No generator <math>w_i</math> normalizes <math>B</math>
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* 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
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* 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
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* the rank of the Tits system is defined to be <math>|S|</math>
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===why do we care?===
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* <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
  
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* <math>G^{u,v} =BuB\cap B_{-}vB_{-}</math>
 
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* <math>G=\cup_{u,v\in W\times W} G^{u,v}</math> (disjoint union)
 
  
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==Bruhat decomposition theorem==
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;thm
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Let <math>G</math> be a group with a <math>BN</math>-pair. Then
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:<math>
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G=BWB
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</math>
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or,
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:<math>
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G=\cup_{w\in W}BwB
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</math>
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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>
  
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==realization of finite type cluster algebra==
 
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* Yang, Shih-Wei, 와/과Andrei Zelevinsky. 2008. “Cluster algebras of finite type via Coxeter elements and principal minors”. <em>0804.3303</em> (4월 21). http://arxiv.org/abs/0804.3303.
 
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* <math>\mathbb{C}[L^{c,c^{-1}}]</math> is a cluster algebra of finite type. It has the same type as Cartan matrix.
 
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===type A_{n}===
 
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* (i) inite seed is given by $x=(x_{[1,1]},\cdots,x_{[1,n]})$, $y=(y_1,\cdots,y_n)$, $B=B(C)$
 
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* (ii) The set of cluster variables is \{x_{[i,j]}|1\leq i\leq j\leq n \}
 
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* (iii) The exchange relations
 
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$$x_{[i,k]}x_{[j,l]} = y_{j-1}y_{j}\cdots y_{k} x_{[i,j-2]}jx_{[i,j-2]}+x_{[i,l]}x_{[j,l]}$$ for $1\leq i\leq j-1\leq k\leq l-1\leq n$
 
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* remark : $x_{[i,j]}$ corresponds to the diagonal between i and j in the triangulation of regular $(n+3)$-gon
 
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==example==
 
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* [[double Bruhat cell example]]
 
  
  
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==related items==
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* [[Cartan decomposition of general linear groups]]
  
 
==computational resource==
 
==computational resource==
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxZzFwSzhRYnRHalE/edit
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxZzFwSzhRYnRHalE/edit
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==related items==
 
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* [[Total positivity]]
 
  
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==encyclopedia==
 
==encyclopedia==
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* http://en.wikipedia.org/wiki/(B,_N)_pair
 
* http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group
 
* http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group
 
* http://eom.springer.de/b/b017690.htm
 
* http://eom.springer.de/b/b017690.htm
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==expositions==
 
==expositions==
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* Lusztig, G. 2010. “Bruhat Decomposition and Applications.” arXiv:1006.5004 [math] (June 25). http://arxiv.org/abs/1006.5004.
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* http://www-math.mit.edu/~gyuri/papers/bru1.pdf
 
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* http://pages.uoregon.edu/dmoseley/talks/
 
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** [http://pages.uoregon.edu/dmoseley/talks/Lecture14.pdf Double Bruhat Cells]
 
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** [http://pages.uoregon.edu/dmoseley/talks/Lecture15.pdf Cluster Structures on Double Bruhat Cells]
 
 
* http://math.ucr.edu/home/baez/week186.html
 
* http://math.ucr.edu/home/baez/week186.html
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* http://www.math.harvard.edu/~ryanr/bruhat_row-reduction.pdf
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* [http://www.ryancreich.info/bruhat_row-reduction.pdf Bruhat decomposition via row reduction]
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==articles==
 
==articles==
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* Yang, Shih-Wei, 와/과Andrei Zelevinsky. 2008. “Cluster algebras of finite type via Coxeter elements and principal minors”. <em>0804.3303</em> (4월 21). http://arxiv.org/abs/0804.3303.<br>
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* '''[C1]''' Chevalley, C. 1955. “Sur Certains Groupes Simples.” The Tohoku Mathematical Journal. Second Series 7: 14–66.
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* Bruhat, Fran\ccois. 1956. “Sur Les Représentations Induites Des Groupes de Lie.” Bulletin de La Société Mathématique de France 84: 97–205.
  
  
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* http://mathoverflow.net/questions/15438/a-slick-proof-of-the-bruhat-decomposition-for-gl-nk
 
* 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/28569/is-there-a-morse-theory-proof-of-the-bruhat-decomposition
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* http://mathoverflow.net/questions/168033/coxeter-groups-parabolic-subgroups/168035#168035
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* http://mathoverflow.net/questions/188920/closure-relations-between-bruhat-cells-on-the-flag-variety/190961#190961
  
 
[[분류:개인노트]]
 
[[분류:개인노트]]
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[[분류:math and physics]]
 
[[분류:math and physics]]
 
[[분류:math]]
 
[[분류:math]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q4978699 Q4978699]
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===Spacy 패턴 목록===
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* [{'LOWER': 'bruhat'}, {'LEMMA': 'decomposition'}]

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:
  1. \(G\) is generated by \(B\) and \(N\)
  2. The intersection, \(T\), of \(B\) and \(N\) is a normal subgroup of N
  3. 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 \(I\)
  4. 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 \(BwB\)
  5. 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


related items

computational resource



encyclopedia


expositions


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

Spacy 패턴 목록

  • [{'LOWER': 'bruhat'}, {'LEMMA': 'decomposition'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Bruhat_decomposition&oldid=51800"