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:
“The only coöperation which is commonly possible is exceedingly partial and superficial; and what little true coöperation there is, is as if it were not, being a harmony inaudible to men. If a man has faith, he will coöperate with equal faith everywhere; if he has not faith, he will continue to live like the rest of the world, whatever company he is joined to.”
—Henry David Thoreau (1817–1862)
““Justice” was done, and the President of the Immortals, in Æschylean phrase, had ended his sport with Tess. And the d’Urberville knights and dames slept on in their tombs unknowing. The two speechless gazers bent themselves down to the earth, as if in prayer, and remained thus a long time, absolutely motionless: the flag continued to wave silently. As soon as they had strength they arose, joined hands again, and went on.
The End”
—Thomas Hardy (1840–1928)
“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.