nLab Bruhat decomposition

Definition

Given a sequence of inclusions

$$T \subset B \subset G$$

where

• $G$ is a connected complex reductive algebraic group

• $B$ is a Borel subgroup

• $T$ is a maximal torus;

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

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

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

into Schubert varieties

$$X_w = \overline{B w B} \subset G/B$$

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

Related concepts

• Schubert calculus

References

Named after François Bruhat.

See also:

• Daniel Bump, The Bruhat decomposition, chapter 27 in: Lie groups, Graduate Texts in Mathematics 225, Springer (2013) [doi:10.1007/978-1-4614-8024-2, pdf]

• Wikipedia, Bruhat decomposition