Flag manifold

Flag manifold

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a generalized flag variety (or simply flag variety) is a homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...

 whose points are flags in a finite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 V over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

, 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
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

 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
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...

. 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
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...

 of parabolic subgroups of a semisimple
Semisimple algebraic group
In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.- Definition :...

 algebraic
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...

 or Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 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
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

 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 the compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 flat model spaces for Cartan geometries of parabolic type, and are homogeneous Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

s under any maximal compact subgroup of G.

Flag manifolds can be symmetric space
Symmetric space
A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...

s. Over the complex numbers, the corresponding flag manifolds are the Hermitian symmetric space
Hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian...

s. 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


A flag in a finite dimensional vector space V over a field F is an increasing sequence of subspace
Subspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...

s, where "increasing" means each is a proper subspace of the next (see filtration
Filtration (abstract algebra)
In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj...

):
If we write the dim Vi = di then we have
where n is the dimension of V. Hence, we must have kn. A flag is called a complete flag if di = i, otherwise it is called a partial flag. The signature of the flag is the sequence (d1, … dk).

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
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, 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
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

 acts
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 transitively on the set of all complete flags.

Fix an ordered basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

 for V, identifying it with Fn, whose general 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
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 of nonsingular upper triangular matrices, which we denote by Bn. The complete flag variety can therefore be written as a homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...

 GL(n,F) / Bn, which shows in particular that it has dimension n(n−1)/2 over F.

Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

 SL(n,F) of matrices with determinant one, which is a semisimple algebraic group; the set of upper triangular matrices of determinant one is a Borel subgroup
Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...

.

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
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...


where U(n) is the unitary group
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...

 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).

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
Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

 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
Block matrix
In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...

 upper triangular matrices, where the dimensions of the blocks are ni := didi−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.

Generalization to semisimple groups


The upper triangular matrices of determinant one are a Borel subgroup of SL(n,F), and hence the stabilizers of partial flags are parabolic subgroups. Furthermore, a partial flag is determined by the parabolic subgroup which stabilizes it, and partial flags belong to the same flag variety precisely when the corresponding parabolic subgroups are conjugate.

Hence, more generally, if G is a semisimple algebraic
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...

 or Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

, then a (generalized) flag variety for G is a conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...

 of parabolic subgroups of G. It is therefore isomorphic, as a homogeneous space, to G/P where P is a parabolic subgroup of G. The correspondence between parabolic subgroups and generalized flag varieties allows each to be understood in terms of the other.

The extension of the terminology "flag variety" is reasonable, because points of G/P can still be described using flags. When G is a classical group, such as a symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...

 or orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

, this is particularly transparent. If (V, ω) is a symplectic vector space
Symplectic vector space
In mathematics, a symplectic vector space is a vector space V equipped with a bilinear form ω : V × V → R that is...

 then a partial flag in V is isotropic if the symplectic form vanishes on proper subspaces of V in the flag. The stabilizer of an isotropic flag is a parabolic subgroup of the symplectic group Sp(V,ω). For orthogonal groups there is a similar picture, with a couple of complications. First, if F is not algebraically closed, then isotropic subspaces may not exist: for a general theory, one needs to use the split orthogonal groups. Second, for vector spaces of even dimension 2m, isotropic subspaces of dimension m come in two flavours ("self-dual" and "anti-self-dual") and one needs to distinguish these to obtain a homogeneous space.

Highest weight orbits and homogeneous projective varieties


If G is a semisimple algebraic group (or Lie group) and V is a (finite dimensional) highest weight representation of G, then the highest weight space is a point in the projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

 P(V) and its orbit under the action of G is a projective algebraic variety. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for G arises in this way.

Armand Borel
Armand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...

 showed that this characterizes the flag varieties of a general semisimple algebraic group G: they are precisely the complete homogeneous spaces of G, or equivalently (in this context), the projective G-varieties.

Symmetric spaces


Let G be a semisimple Lie group with maximal compact subgroup K. Then K acts transitively on any conjugacy class of parabolic subgroups, and hence the generalized flag variety G/P is a compact homogeneous Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 K/(KP) with isometry group K. Furthermore, if G is a complex Lie group, G/P is a homogeneous Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

.

Turning this around, the Riemannian homogeneous spaces
M = K/(KP)


admit a strictly larger Lie group of transformations, namely G. Specializing to the case that M is a symmetric space
Symmetric space
A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...

, this observation yields all symmetric spaces admitting such a larger symmetry group, and these spaces have been classified by Kobayashi and Nagano.

If G is a complex Lie group, the symmetric spaces M arising in this way are the compact Hermitian symmetric space
Hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian...

s: K is the isometry group, and G is the biholomorphism group of M.

Over the real numbers, a real flag manifold is also called an R-space, and the R-spaces which are Riemannian symmetric spaces under K are known as symmetric R-spaces. The symmetric R-spaces which are not Hermitian symmetric are obtained by taking G to be a real form of the biholomorphism group Gc of a Hermitian symmetric space Gc/Pc such that P := PcG is a parabolic subgroup of G. Examples include projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

s (with G the group of projective transformations) and sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

s (with G the group of conformal transformations).