"Bruhat decomposition"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
imported>Pythagoras0
1번째 줄: 1번째 줄:
 
==introduction==
 
==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
 
* 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
10번째 줄: 10번째 줄:
  
 
==example : general linear group==
 
==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==
 
==(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?===
 
===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
 
* 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
 
* BN-pairs can be used to prove that most groups of Lie type are simple
35번째 줄: 35번째 줄:
 
==Bruhat decomposition theorem==
 
==Bruhat decomposition theorem==
 
;thm
 
;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
 
G=BWB
−
$$
+
</math>
 
or,  
 
or,  
−
$$
+
:<math>
 
G=\cup_{w\in W}BwB
 
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>
  
  

2020년 11월 16일 (월) 04:30 판

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)

원본 주소 "https://wiki.mathnt.net/index.php?title=Bruhat_decomposition&oldid=44941"