# nLab Bruhat decomposition

Contents

### Context

#### Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Definition

Given a sequence of inclusions

$T \subset B \subset G$

where

this induces

• the Weyl group $W_0 = N(T)/T$;

• the character lattice $\mathfrak{h}_{\mathbb{Z}}^\ast = Hom(T, \mathbb{C}^\times)$;

• the cocharacter lattice $\mathfrak{h}_{\mathbb{C}} = Hom(\mathbb{C}^\times, T)$.

• a standard parabolic subgroup of $G$ is a subgroup $P_J$ including $B$ such that $G/P$ is a projective variety;

parabolic subgroup is one conjugate to the standard parabolic subgroup.

• the flag variety $G/B$;

• the partial flag varieties $G/P_J$

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

$G = \underset{w \in W_0}{\coprod} B w B$
$G = \underset{u \in W^J}{\coprod} B u P_j$

with

• $W_J \coloneqq \{v \in W_0 | v T \subset P_J\}$

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

$X_w = \overline{B w B} \subset G/B$
$X_u^J = \overline{B u P_J} \subset G/P_J \,.$

## References

Named after François Bruhat.