Flag manifold

In mathematics, a flag is an increasing sequence of subspaces of a vector space. "Increasing" means each is a subspace of the next. According to basic results of linear algebra, any two flags of a finite-dimensional vector space V, in which the dimensions are the same, are no different from each other from a geometric point of view. Therefore if we specify a sequence of dimensions 0 < d₁ < d₂ < ... < d_k < n, where n is the dimension of V, we can consider a homogeneous space F(d₁, d₂, ... , d_k) = H\\G 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. We call F(d₁, d₂, ... , d_k) a flag manifold.

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. Just as in that case, when K is the real or complex field we can also consider the flag manifolds as coset spaces for Lie groups - in more than one way, since we can for example use orthogonal groups in the real case rather than the general linear group.

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.

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 stabiliser of a complete flag, where d_i = i and k=n-1. The corresponding manifold is often called a full flag manifold; the others, with some gap in dimensions, are then referred to as partial flag manifolds.

The case in which we take d_i = i and k < n-1 is also called a Stiefel manifold: it corresponds, relative to the Grassmannian G_k,n, to taking additionally a basis (up to scalar multiples) of the k-dimensional subspace. There is an important mapping from a Stiefel manifold to the Grassmannian forgetting the extra information. It is closely related to the principal bundle for the associated universal or tautological bundle on the Grassmannian.

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'.