Flag variety - Knowlpedia

Definition

Let $G$ be a connected reductive algebraic group over a field $k$ , and let $P\subseteq G$ be a parabolic subgroup. The quotient $G/P$ is a flag variety. When $P=B$ is a Borel subgroup, $G/B$ is the complete flag variety.

For $G=GL_n$ , these varieties parametrize chains

0\subset V_{d_1}\subset\cdots\subset V_{d_r}\subset k^n, \qquad \dim V_{d_i}=d_i.

The complete flag variety uses every dimension $1,\ldots,n-1$ .

Geometry

Flag varieties are smooth and projective. The group $G$ acts transitively, and the stabilizer of the base flag is $P$ . Their Bruhat decompositions are indexed by suitable cosets in the Weyl group.

The rank-one case $SL_2/B\simeq\mathbb P^1$ is the simplest example.

References

Claude Chevalley, “Sur les décompositions cellulaires des espaces $G/B$ ,” in Algebraic Groups and Their Generalizations: Classical Methods, Proc. Sympos. Pure Math. 56 (1994), 1–23 (manuscript originally circulated in 1958).