Grassmannian


In mathematics, a Grassmannian, also known as a Grassmann manifold is a differentiable manifold that parameterizes the set of all -dimensional linear subspaces of an -dimensional vector space over a field that has a differentiable structure. For example, the Grassmannian is the space of lines through the origin in, so it is the same as the projective space of one dimension lower than.
When is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension. In general they have the structure of a nonsingular projective algebraic variety. The Grassmannian is named for the German polymath, linguist and mathematician Hermann Grassmann, who introduced the concept to mathematics.

History

The earliest work on a non-trivial Grassmannian was by Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to, parameterizing them by what are now called Plücker coordinates. Hermann Grassmann later generalized the concept.
Notations for Grassmannians vary between authors; they include,,, to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space.

Motivation

By giving a collection of subspaces of a vector space a topological structure, it is possible to talk about a continuous choice of subspaces or open and closed collections of subspaces. Giving them the further structure of a differentiable manifold enables smooth choices of subspace.
A natural example comes from tangent bundles of smooth manifolds embedded in a Euclidean space. Given a manifold of dimension embedded in. At each point, the tangent space to can be considered as a subspace of the tangent space of, which is also just. The map assigning to its tangent space defines a map from to.
This can be extended to all vector bundles over a manifold, so that every vector bundle generates a continuous map from to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. The properties of vector bundles are thus related to the properties of the corresponding maps. In particular vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopy relies on a notion of continuity, and hence a topology.

Low dimensions

For, the Grassmannian is the space of lines through the origin in -space, so it is the same as the projective space of dimensions.
For, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the single line through the origin that is perpendicular to that plane ; hence the spaces,, and may all be identified with each other.
The simplest Grassmannian that is not a projective space is.

Differentiable manifold

To endow with the structure of a differentiable manifold, a basis for must be chosen. This is equivalent to identifying with, with the standard basis denoted, viewed as column vectors. Then for any -dimensional subspace, viewed as an element of, a basis can be found consisting of linearly independent column vectors. The homogeneous coordinates of the element consist of the elements of the maximal rank rectangular matrix whose -th column vector is,. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices and represent the same element if and only if
for some element of the general linear group of invertible matrices with entries in. This defines an equivalence relation between matrices of rank, for which the equivalence classes are denoted.
A coordinate atlas ensures that for any homogeneous coordinate matrix, elementary column operations can be applied to obtain its reduced column echelon form. If the first rows of are linearly independent, the result has the form
and the affine coordinate matrix
with entries determines. In general, the first rows need not be independent, but since has maximal rank, an ordered set of integers exists such that the submatrix whose rows are the -th rows of is nonsingular. Column operations can reduce this submatrix to the identity matrix, and the remaining entries uniquely determine. This gives the following definition:
For each ordered set of integers, let a set of elements exists for which, for any choice of homogeneous coordinate matrix, the submatrix whose -th row is the -th row of is nonsingular. The affine coordinate functions on are then defined as the entries of the matrix whose rows are those of the matrix complementary to, written in the same order. The choice of homogeneous coordinate matrix in representing the element does not affect the values of the affine coordinate matrix representing on the coordinate neighbourhood. Moreover, the coordinate matrices may take arbitrary values, and they define a diffeomorphism from to the space of -valued matrices. This can be denoted by
the homogeneous coordinate matrix having the identity matrix as the submatrix with rows and the affine coordinate matrix in the consecutive complementary rows. On the overlap between any two such coordinate neighborhoods, the affine coordinate matrix values and are related by the transition relations
where both and are invertible. This may equivalently be written as
where is the invertible matrix whose th row is the th row of. The transition functions are therefore rational in the matrix elements of, and gives an atlas for as a differentiable manifold and also as an algebraic variety.

Orthogonal projections

An alternative way to define a real or complex Grassmannian as a manifold is to view it as a set of orthogonal projection operators. For this, a positive definite real or Hermitian inner product on can be chosen, depending on whether is real or complex. A -dimensional subspace determines a unique orthogonal projection operator whose image is by splitting into the orthogonal direct sum
of and its orthogonal complement and defining
Conversely, every projection operator of rank defines a subspace as its image. Since the rank of an orthogonal projection operator equals its trace, we can identify the Grassmann manifold with the set of rank orthogonal projection operators :
In particular, taking or gives completely explicit equations for embedding the Grassmannians, in the space of real or complex matrices,, respectively.
Since this defines the Grassmannian as a closed subset of the sphere the Grassmannian is a compact Hausdorff space. This construction also turns the Grassmannian into a metric space with metric
for any pair of -dimensional subspaces, where denotes the operator norm. The exact inner product used does not matter, because a different inner product gives an equivalent norm on, and hence an equivalent metric.
For the case of real or complex Grassmannians, the following is an equivalent way to express the above construction in terms of matrices.

Affine algebraic varieties

Let denote the space of real matrices and the subset of matrices that satisfy the three conditions:
  • is a projection operator:.
  • is symmetric:.
  • has trace.
A bijective correspondence exists between and the Grassmannian of -dimensional subspaces of given by sending to the -dimensional subspace of spanned by its columns and, conversely, sending any element to the projection matrix
where is any orthonormal basis for, viewed as real component column vectors.
An analogous construction applies to the complex Grassmannian, identifying it bijectively with the subset of complex matrices satisfying
where the self-adjointness is with respect to the Hermitian inner product in which the standard basis vectors are orthonomal. The formula for the orthogonal projection matrix onto the complex -dimensional subspace spanned by the orthonormal basis vectors is

Homogeneous space

The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space. The general linear group acts transitively on the -dimensional subspaces of. Therefore, by choosing a subspace of dimension, any element can be expressed as
for some group element, where is determined only up to right multiplication by elements of the stabilizer of :
under the -action.
can be identified with the quotient space
of left cosets of.
If the underlying field or and is considered as a Lie group, this construction makes the Grassmannian a smooth manifold under the quotient structure. More generally, over a ground field, the group is an algebraic group, and this construction shows that the Grassmannian is a non-singular algebraic variety. It follows from the existence of the Plücker embedding that the Grassmannian is complete as an algebraic variety. In particular, is a parabolic subgroup of.
Over or it becomes possible to use smaller groups in this construction. To do this over, a Euclidean inner product can be fixed on. The real orthogonal group acts transitively on the set of -dimensional subspaces and the stabiliser of a -space is
where is the orthogonal complement of in. This gives an identification as the homogeneous space
taking and gives the isomorphism
Over, choosing an Hermitian inner product let the unitary group act transitively, and analogously
or, for and,
In particular, this shows that the Grassmannian is compact, and of dimension.

Scheme

In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.

Representable functor

If is a quasi-coherent sheaf on a scheme for a positive integer, then to each -scheme, the Grassmannian functor associates the set of quotient modules of
locally free of rank on. We denote this set by.
This functor is representable by a separated -scheme. The latter is projective if is finitely generated. When is the spectrum of a field, then the sheaf is given by a vector space and the usual Grassmannian variety of the dual space of can be recovered, namely:. By construction, the Grassmannian scheme is compatible with base changes: for any -scheme, giving the canonical isomorphism
In particular, for any point of, the canonical morphism
induces an isomorphism from the fiber to the usual Grassmannian over the residue field.