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. Math sucks. ... In mathematics, a flag is an increasing sequence of subspaces of a vector space. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... This article is about algebraic varieties. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...

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.

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. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... This article is about the mathematical concept. ...

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. In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... This article is about mathematical concept. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...

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). In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

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. In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ...

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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... This article is about algebraic varieties. ... In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ...

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. In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...

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. In mathematics, a Borel subgroup (named after Armand Borel) of an algebraic group G is a maximal solvable subgroup. ... In mathematics, 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 reveals many important features of a groups structure. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In mathematics, a Borel subgroup (named after Armand Borel) of an algebraic group G is a maximal solvable subgroup. ...

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'. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K containing the eigenvalues of M, to what extent can M... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ...