Bruhat ordering

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

• https://drive.google.com/file/d/0B8XXo8Tve1cxdko1Z0hPZUNhZTQ/view
• http://math.univ-lyon1.fr/homes-www/ducloux/coxeter/coxeter3/english/bruhat_e.html