nLab
Bruhat decomposition

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Context

Group Theory

Geometry

Contents

Definition

Given a sequence of inclusions

TBG T \subset B \subset G

where

this induces

A Bruhat decomposition is, if it exists, a coproduct decomposition into a disjoint union of double cosets

G=wW 0BwB G = \underset{w \in W_0}{\coprod} B w B
G=uW JBuP j G = \underset{u \in W^J}{\coprod} B u P_j

with

  • W J{vW 0|vTP J}W_J \coloneqq \{v \in W_0 | v T \subset P_J\}

  • W J{cosetrepresentativesuofcosetsinW 0/W J}W^J \coloneqq \{coset\; representatives\; u \; of \; cosets \; in W_0/W_J\}

into Schubert varieties

X w=BwB¯G/B X_w = \overline{B w B} \subset G/B
X u J=BuP J¯G/P J. X_u^J = \overline{B u P_J} \subset G/P_J \,.

References

Last revised on August 2, 2017 at 08:40:28. See the history of this page for a list of all contributions to it.