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.
|Date of creation||2013-03-22 13:27:56|
|Last modified on||2013-03-22 13:27:56|
|Last modified by||bwebste (988)|
|Defines||complete flag variety|
|Defines||partial flag variety|