Flag manifold

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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.

Flag manifolds are called complete or partial according to whether one considers complete or partial flags. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration.

Contents 1 Generalization 2 As a homogeneous space 3 As algebraic varieties 4 Subgroups of the general linear group 5 Topology

[edit] Generalization

If G is a semisimple algebraic group then its complete flag variety is the homogeneous space G/B and partial flag varieties have the form G/P, where B is a Borel subgroup and P is a parabolic subgroup of G. The case of the flag manifold corresponds to taking G to be general (or special) linear group. For a classical group G acting on a formed space by isometries, its partial flag varieties can be described in terms of flags in the space satisfying additional conditions (e.g. isotropic or self-dual). Armand Borel found an elegant characterization of flag varieties for a general semisimple G: they are complete homogeneous spaces of G, or projective G-varieties, which in this situation amounts to the same thing.

[edit] 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.

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 B_n. The flag variety can therefore be written as a homogeneous space GL(n) / B_n. This shows that the dimension of the flag variety is

n(n − 1)/2.

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).

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

0 = d₀ < d₁ < d₂ < ... < d_k < d_k+1 = n,

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

F(d₁, d₂, ..., d_k) = G/H

of all flags of that type. Here H must therefore be taken as the stabilizer of one such flag given by subspaces V_i of dimension d_i, 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 d_i − d_i−1.

[edit] As algebraic varieties

This much works over any field K. The flag manifold is an algebraic variety over K; which turns out to be a projective variety. These varieties therefore include the Grassmannians, which are the special case where k = 1: i.e. we take just one intermediate space V₁.

To look more closely at the stabilizer H, one can take a standard basis e₁, ..., e_n, and V_i to be spanned by the first d_i of them. Then as a matrix group H has a definite block structure; in fact the various H correspond to the various ways of considering what 'below the diagonal' means in block matrix terms, by demanding entries that are 0 there. This can be applied, for example, to count flags over finite fields, as is done on the general linear group page.

[edit] Subgroups of the general linear group

It also gives a survey of all the parabolic subgroups of the general linear group, up to conjugacy. That is, in this case the abstract algebraic group theory of parabolic subgroups (those containing a Borel subgroup) can be read off from the flag manifolds, considered collectively. The subgroup of upper triangular matrices is in this case a Borel subgroup: it corresponds to the stabilizer of a complete flag.

[edit] Topology

It is also possible to read off topological information about the groups H. From the point of view of homotopy theory, the unipotent part of the Jordan normal forms is a contractible factor in a direct product decomposition, and so makes no contribution. In this way one can to read off topological principles for vector bundles. Reduction of the structure group of such a bundle to one of the groups H implies the existence of sub-bundles. The obstructions will lie in the diagonal block parts, not in the above-diagonal part. For example the reduction to upper-triangular form implies reduction to diagonal form, and so sum of line bundles. This gives rise therefore to general 'splitting principles'.