Flag manifold

In mathematics, a flag manifold (or flag variety) is the set of all flags in a finite-dimensional vector space V. The flag variety on V is naturally a projective variety. If the base field K is the real or complex numbers then the flag variety has a natural manifold structure turning it into a smooth or complex manifold.

Subgroups of the general linear group

It also gives a survey of all the parabolic subgroups of the general linear group, up to conjugacy. That is, in this case the abstract algebraic group theory of parabolic subgroups (those containing a Borel subgroup) can be read off from the flag manifolds, considered collectively. The subgroup of upper triangular matrices is in this case a Borel subgroup: it corresponds to the stabilizer of a complete flag.

Related Topics:
Parabolic subgroup - Conjugacy - Algebraic group - Borel subgroup

~ ~ ~ ~ ~ ~ ~ ~ ~ ~