Flag manifold

In mathematics, a flag manifold (or flag variety) is the set of all flags in a finite-dimensional vector space V. The flag variety on V is naturally a projective variety. If the base field K is the real or complex numbers then the flag variety has a natural manifold structure turning it into a smooth or complex manifold.

As a homogeneous space

According to basic results of linear algebra, any two (complete) flags of an n-dimensional vector space V are no different from each other from a geometric point of view. That is to say, the general linear group acts transitively on the set of all flags.

Related Topics:
Linear algebra - General linear group - Acts

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Fix an ordered basis for V. The standard flag associated with this basis is the one where the ith subspace is spanned by the first i vectors of the basis. Relative to this basis, the stabilizer of the standard flag is the group of nonsingular upper triangular matrices, which we denote by Bn. The flag variety can therefore be written as a homogeneous space GLn / Bn. This shows that the dimension of the flag variety is n(n−1)/2.

Related Topics:
Basis - Stabilizer - Group - Upper triangular matrices - Homogeneous space

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Flag varieties can often be considered as homogeneous spaces in more than one way. For instance, when K is the field of real numbers the orthogonal group O(n) acts transitively on the set of all flags (with the stabilizer subgroup H equal to the diagonal subgroup).

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To handle partial flag varieties we need to specify a sequence of dimensions

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:0 = d0 < d1 < d2 < ... < dk < dk+1 = n,

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where n is the dimension of V. A complete flag is the special case of di = i and k = n−1. We can consider a homogeneous space

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:F(d1, d2, ..., dk) = G/H

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of all flags of that type. Here H must therefore be taken as the stabilizer of one such flag given by subspaces Vi of dimension di, that are nested. For instance, if G is the general linear group, the H can be taken to be the group of nonsingular block upper triangular matrices, where the dimensions of the blocks are di − di−1.

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