Bruhat ordering
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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'