"Bruhat ordering"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
imported>Pythagoras0 |
||
| 1번째 줄: | 1번째 줄: | ||
| + | ==introduction== | ||
* Let $W$ be a Coxeter group | * Let $W$ be a Coxeter group | ||
| 10번째 줄: | 11번째 줄: | ||
;thm | ;thm | ||
Given $x,y\in W$, we have $x<y$ 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 $x<y$ 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). | ||
==computational resource== | ==computational resource== | ||
* https://drive.google.com/file/d/0B8XXo8Tve1cxdko1Z0hPZUNhZTQ/view | * https://drive.google.com/file/d/0B8XXo8Tve1cxdko1Z0hPZUNhZTQ/view | ||
2016년 5월 2일 (월) 07:07 판
introduction
- Let $W$ be a Coxeter group
- def (Bruhat ordering)
Define a partial order on the elements of $W$ as follows :
$u < v$ whenever $u = t v$ for some reflection $t$ and $\ell(u) < \ell(v)$.
- thm
Given $x,y\in W$, we have $x<y$ 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).