"Bruhat ordering"의 두 판 사이의 차이

2020년 11월 16일 (월) 10:03 기준 최신판

introduction

Let \(W\) be a Coxeter group

def (Bruhat ordering)

Define a partial order on the elements of \(W\) as follows :

Write \(w'\xrightarrow{t} w\) whenever \(w = w' t\) for some reflection \(t\) and \(\ell(w') < \ell(w)\). Define \(w'<w\) if there is a sequence \(w'=w_0\to w_1\to \cdots \to w_n=w\). Extend this relation to a partial ordering of \(W\). (reflexive, antisymmetric, transitive)

example : http://groupprops.subwiki.org/wiki/File:Bruhatons3.png

thm

Given \(x,y\in W\), we have \(x\le y\) in the Bruhat order if and only if there is a reduced expression \(y=s_{i_1}s_{i_2}\cdots s_{i_k}\) such that \(x\) can be written as a product of some of the \(s_{i_j}\) in the same order as they appear in \(y\).

history

The Bruhat order on the Schubert varieties of a flag manifold or Grassmannian was first studied by Ehresmann (1934), and the analogue for more general semisimple algebraic groups was studied by Chevalley (1958).

memo

See also Chapter 8 of Humphereys' 'Reflection groups and Coxeter groups'

related items

Flag manifold and flag variety

computational resource

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* Let $W$ be a Coxeter group
+* Let <math>W</math> be a Coxeter group
 ;def (Bruhat ordering)
-Define a partial order on the elements of $W$ as follows :
+Define a partial order on the elements of <math>W</math> as follows :
-Write $w'\xrightarrow{t} w$ whenever $w = w' t$ for some reflection $t$ and $\ell(w') < \ell(w)$. Define $w'<w$ if there is a sequence $w'=w_0\to w_1\to \cdots \to w_n=w$. Extend this relation to a partial ordering of $W$. (reflexive, antisymmetric, transitive)
+Write <math>w'\xrightarrow{t} w</math> whenever <math>w = w' t</math> for some reflection <math>t</math> and <math>\ell(w') < \ell(w)</math>. Define <math>w'<w</math> if there is a sequence <math>w'=w_0\to w_1\to \cdots \to w_n=w</math>. Extend this relation to a partial ordering of <math>W</math>. (reflexive, antisymmetric, transitive)
 * example : http://groupprops.subwiki.org/wiki/File:Bruhatons3.png
 ;thm
-Given $x,y\in W$, we have $x\le y$ in the Bruhat order if and only if there is a reduced expression $y=s_{i_1}s_{i_2}\cdots s_{i_k}$ such that $x$ can be written as a product of some of the $s_{i_j}$ in the same order as they appear in $y$.
+Given <math>x,y\in W</math>, we have <math>x\le y</math> in the Bruhat order if and only if there is a reduced expression <math>y=s_{i_1}s_{i_2}\cdots s_{i_k}</math> such that <math>x</math> can be written as a product of some of the <math>s_{i_j}</math> in the same order as they appear in <math>y</math>.
 ==history==