Partial Flag Varieties
The partial flag variety
is the space of all flags of signature (d1, d2, … dk) in a vector space V of dimension n = dk over F. The complete flag variety is the special case that di = i for all i. When k=2, this is a Grassmannian of d1-dimensional subspaces of V.
This is a homogeneous space for the general linear group G of V over F. To be explicit, take V = Fn so that G = GL(n,F). The stabilizer of a flag of nested subspaces Vi of dimension di can be taken to be the group of nonsingular block upper triangular matrices, where the dimensions of the blocks are ni := di − di−1 (with d0 = 0).
Restricting to matrices of determinant one, this is a parabolic subgroup P of SL(n,F), and thus the partial flag variety is isomorphic to the homogeneous space SL(n,F)/P.
If F is the real or complex numbers, then an inner product can be used to split any flag into a direct sum, and so the partial flag variety is also isomorphic to the homogeneous space
in the complex case, or
in the real case.
Read more about this topic: Generalized Flag Variety
Famous quotes containing the words partial, flag and/or varieties:
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“Now there are varieties of gifts, but the same Spirit; and there are varieties of services, but the same Lord; and there are varieties of activities, but it is the same God who activates all of them in everyone.”
—Bible: New Testament, 1 Corinthians 12:4-6.