Generalized flag variety

In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties.

Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags.

The most general concept of a generalized flag variety is a conjugacy class of parabolic subgroups of a semisimple algebraic or Lie group G: G acts transitively on such a conjugacy class by conjugation, and the stabilizer of a parabolic P is P itself, so that the generalized flag variety is isomorphic to G/P. It may also be realised as the orbit of a highest weight space in a projectivized representation of G. In the algebraic setting, generalized flag varieties are precisely the homogeneous spaces for G which are complete as algebraic varieties. In the smooth setting, generalized flag manifolds are compact, and are homogeneous Riemannian manifolds under any maximal compact subgroup of G.

Flag manifolds can be symmetric spaces. Over the complex numbers, the corresponding flag manifolds are the Hermitian symmetric spaces. Over the real numbers, an R-space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric R-spaces.

Flags in a vector space

Main article: flag (linear algebra)

A flag in a finite dimensional vector space V over a field F is an increasing sequence of subspaces, where "increasing" means each is a proper subspace of the next (see filtration):

$\{0\}=V_{0}\subset V_{1}\subset V_{2}\subset \cdots \subset V_{k}=V.$

If we write the dim V_i = d_i then we have

$0=d_{0}<d_{1}<d_{2}<\cdots <d_{k}=n,$

where n is the dimension of V. Hence, we must have k ≤ n. A flag is called a complete flag if d_i = i, otherwise it is called a partial flag. The signature of the flag is the sequence (d₁, … d_k).

A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.

Prototype: the complete flag variety

According to basic results of linear algebra, any two complete flags in an n-dimensional vector space V over a field F 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 complete flags.

Fix an ordered basis for V, identifying it with Fⁿ, whose special linear group is the group GL(n,F) of n × n matrices. The standard flag associated with this basis is the one where the i th 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 complete flag variety can therefore be written as a homogeneous space GL(n,F) / B_n, which shows in particular that it has dimension n(n−1)/2 over F.

If the field F is the real or complex numbers we can introduce an inner product on V such that the chosen basis is orthonormal. Any complete flag then splits into a direct sum of one dimensional subspaces by taking orthogonal complements. It follows that the complete flag manifold over the complex numbers is the homogeneous space

$U(n)/T^{n}$

where U(n) is the unitary group and Tⁿ is the n-torus of diagonal unitary matrices. There is a similar description over the real numbers with U(n) replaced by the orthogonal group O(n), and Tⁿ by the diagonal orthogonal matrices (which have diagonal entries ±1).

Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the special linear group SL(n,F) of matrices with determinant one, which is a semisimple algebraic group.

Partial flag varieties

The partial flag variety

$F(d_{1},d_{2},\ldots d_{k},\mathbb {F} )$

is the space of all flags of signature (d₁, d₂, … d_k) in a vector space V of dimension n = d_k over F. The complete flag variety is the special case that d_i = 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 V_i of dimension d_i can be taken to be the group of nonsingular block upper triangular matrices, where the dimensions of the blocks are n_i := d_i − d_i−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

$U(n)/U(n_{1})\times \cdots \times U(n_{k})$

in the complex case, or

$O(n)/O(n_{1})\times \cdots \times O(n_{k})$

in the real case.

Flag varieties for algebraic groups

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 fundamental representation 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.

As algebraic varieties

This much works over any field F. The flag manifold is an algebraic variety over F; 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.

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.

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 generalizations of the splitting principle.

References

• Robert J. Baston and Michael G. Eastwood, The Penrose Transform: its Interaction with Representation Theory, Oxford University Press, 1989.
• Jürgen Berndt, Lie group actions on manifolds, Lecture notes, Tokyo, 2002.
• Jürgen Berndt, Sergio Console and Carlos Olmos, Submanifolds and Holonomy, Chapman & Hall/CRC Press, 2003.
• James E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Springer-Verlag, 1972.
• Michel Brion, Lectures on the geometry of flag varieties, Lecture notes, Varsovie, 2003.