flag variety

Let k be a field, and let V be a vector spaceMathworldPlanetmath over k of dimensionPlanetmathPlanetmathPlanetmath n and choose an increasing sequence 𝐢=(i1,,im), with 1i1<<imn. Then the (partial) flag variety (V,𝐢) associated to this data is the set of all flags {0}V1Vn with dimVj=ij. This has a natural embedding into the product of Grassmannians G(V,i1)×G(V,im), and its image here is closed, making (V,𝐢) into a projective variety over k. If k= these are often called flag manifolds.

The group Sl(V) acts transtively on (V,𝐢), and the stabilizerMathworldPlanetmath of a point is a parabolic subgroup. Thus, as a homogeneous space, (V,𝐢)Sl(V)/P where P is a parabolic subgroup of Sl(V). In particular, the complete flag variety is isomorphicPlanetmathPlanetmathPlanetmath to Sl(V)/B, where B 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