"Bruhat ordering"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 5번째 줄: | 5번째 줄: | ||
Define a partial order on the elements of $W$ as follows : | Define a partial order on the elements of $W$ as follows : | ||
| − | Write $w'\ | + | 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 | * example : http://groupprops.subwiki.org/wiki/File:Bruhatons3.png | ||
| 11번째 줄: | 11번째 줄: | ||
;thm | ;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 $x,y\in W$, 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 $y$. | ||
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==history== | ==history== | ||
2016년 5월 4일 (수) 00:26 판
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)
- 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'