flag variety
Let be a field, and let be a vector space over of dimension
and
choose an increasing sequence , with
. Then the (partial) flag variety
associated to this data is the set of all
flags with . This has a
natural embedding into the product of Grassmannians
, and its image here is closed, making
into a projective variety over . If
these are often called flag manifolds.
The group acts transtively on ,
and the stabilizer of a point is a parabolic subgroup. Thus, as a homogeneous
space, where is a parabolic
subgroup of . In particular, the complete flag variety is
isomorphic
to , where is the Borel subgroup.
Title | flag variety |
---|---|
Canonical name | FlagVariety |
Date of creation | 2013-03-22 13:27:56 |
Last modified on | 2013-03-22 13:27:56 |
Owner | bwebste (988) |
Last modified by | bwebste (988) |
Numerical id | 6 |
Author | bwebste (988) |
Entry type | Definition |
Classification | msc 14M15 |
Synonym | flag manifold |
Defines | complete flag variety |
Defines | partial flag variety |