Grassmannian

In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . 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. In general they have the structure of a smooth algebraic variety, of dimension formula_1
The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general.
Notations vary between authors, with being equivalent to , and some authors using or to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space over an unspecified field.
Motivation.
By giving a collection of subspaces of some vector space a topological structure, it is possible to talk about a continuous choice of subspace or open and closed collections of subspaces; by giving them the structure of a differential manifold one can talk about smooth choices of subspace.
A natural example comes from tangent bundles of smooth manifolds embedded in Euclidean space. Suppose we have a manifold of dimension embedded in . At each point in , the tangent space to can be considered as a subspace of the tangent space of , which is just . The map assigning to its tangent space defines a map from to . (In order to do this, we have to translate the tangent space at each ∈ so that it passes through the origin rather than , and hence defines a -dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.)
This idea can with some effort 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. We then find that the properties of our vector bundles are related to the properties of the corresponding maps viewed as continuous maps. In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopic 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 one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces , , and (the projective plane) may all be identified with each other.
The simplest Grassmannian that is not a projective space is .
The geometric definition of the Grassmannian as a set.
Let be an -dimensional vector space over a field . The Grassmannian is the set of all -dimensional linear subspaces of . The Grassmannian is also denoted or .
The Grassmannian as a differentiable manifold.
To endow the Grassmannian with the structure of a differentiable manifold, choose
a basis for . This is equivalent to identifying it with with the standard basis, denoted formula_2, viewed as column vectors. Then for any -dimensional subspace , viewed as an element of , we may choose a basis consisting of linearly independent column vectors formula_3. The homogeneous coordinates of the element consist of the components of the rectangular matrix of maximal rank whose th column vector is formula_4. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices and formula_5 represent the same element if and only if formula_6 for some element of the general linear group of invertible matrices with entries in .
Now we define a coordinate atlas. For any matrix , we can apply elementary column operations to obtain its reduced column echelon form. If the first rows of are linearly independent, the result will have the form
The matrix determines . In general, the first rows need not be independent, but for any whose rank is formula_8, there exists an ordered set of integers formula_9 such that the submatrix formula_10 consisting of the formula_11-th rows of is nonsingular. We may apply column operations to reduce this submatrix to the identity, and the remaining entries uniquely correspond to . Hence we have the following definition:
For each ordered set of integers formula_9, let formula_13 be the set of formula_14 matrices whose submatrix formula_10 is nonsingular, where the th row of formula_10 is the th row of . The coordinate function on formula_13 is then defined as the map formula_18 that sends to the rectangular matrix whose rows are the rows of the matrix formula_19 complementary to formula_20. The choice of homogeneous coordinate matrix representing the element does not affect the values of the coordinate matrix formula_21 representing on the coordinate neighbourhood formula_13. Moreover, the coordinate matrices formula_23 may take arbitrary values, and they define a diffeomorphism from formula_13 onto the space of -valued matrices.
On the overlap
of any two such coordinate neighborhoods, the coordinate matrix values are related by the transition relation
where both formula_27 and formula_28 are invertible. Hence formula_29 gives an atlas of .
The Grassmannian as a homogeneous space.
The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space. First, recall that the general linear group acts transitively on the -dimensional subspaces of . Therefore, if is the stabilizer of any of the subspaces under this action, we have
If the underlying field is or and is considered as a Lie group, then this construction makes the Grassmannian into a smooth manifold. It also becomes possible to use other groups to make this construction. To do this, fix an inner product on . Over , one replaces by the orthogonal group , and by restricting to orthonormal frames, one gets the identity
In particular, the dimension of the Grassmannian is .
Over , one replaces by the unitary group . This shows that the Grassmannian is compact. These constructions also make the Grassmannian into a metric space: For a subspace of , let be the projection of onto . Then
where denotes the operator norm, is a metric on . The exact inner product used does not matter, because a different inner product will give an equivalent norm on , and so give an equivalent metric.
If the ground field is arbitrary and is considered as an algebraic group, then 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 .
The Grassmannian as a scheme.
In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.
Representable functor.
Let formula_31 be a quasi-coherent sheaf on a scheme . Fix 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 formula_33.
This functor is representable by a separated -scheme formula_34. The latter is projective if formula_31 is finitely generated. When is the spectrum of a field , then the sheaf formula_31 is given by a vector space and we recover the usual Grassmannian variety of the dual space of , namely: .
By construction, the Grassmannian scheme is compatible with base changes: for any -scheme , we have a canonical isomorphism
In particular, for any point of , the canonical morphism , induces an isomorphism from the fiber formula_38 to the usual Grassmannian formula_39 over the residue field .
Universal family.
Since the Grassmannian scheme represents a functor, it comes with a universal object, formula_40, which is an object of
and therefore a quotient module formula_40 of formula_43, locally free of rank over formula_34. The quotient homomorphism induces a closed immersion from the projective bundle formula_45:
For any morphism of -schemes:
this closed immersion induces a closed immersion
Conversely, any such closed immersion comes from a surjective homomorphism of -modules from formula_49 to a locally free module of rank . Therefore, the elements of formula_50 are exactly the projective subbundles of rank in
Under this identification, when is the spectrum of a field and formula_31 is given by a vector space , the set of rational points formula_53 correspond to the projective linear subspaces of dimension in , and the image of formula_54 in
is the set
The Plücker embedding.
The Plücker embedding is a natural embedding of the Grassmannian formula_57 into the projectivization of the exterior algebra :
Suppose that is a -dimensional subspace of the -dimensional vector space . To define formula_59, choose a basis of , and let formula_59 be the wedge product of these basis elements:
A different basis for will give a different wedge product, but the two products will differ only by a non-zero scalar (the determinant of the change of basis matrix). Since the right-hand side takes values in a projective space, formula_62 is well-defined. To see that formula_62 is an embedding, notice that it is possible to recover from formula_62 as the span of the set of all vectors such that formula_65.
Plücker coordinates and the Plücker relations.
The Plücker embedding of the Grassmannian satisfies some very simple quadratic relations called the Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of and give another method of constructing the Grassmannian. To state the Plücker relations, fix a basis of , and let be a -dimensional subspace of with basis }. Let be the coordinates of with respect to the chosen basis of , let formula_66 and let be the columns of formula_67. For any ordered sequence formula_68 of formula_8 positive integers, let formula_70 be the determinant of the formula_71 matrix with columns formula_72. The set formula_73 is called the Plücker coordinates of the element formula_74 of the Grassmannian (with respect to the basis of ). They are the linear coordinates of the image formula_59 of formula_76 under the Plücker map, relative to the basis of the exterior power induced by the basis of .
For any two ordered sequences formula_77 and formula_78 of formula_79 and formula_80 positive integers, respectively, the following homogeneous equations are valid and determine the image of under the Plücker embedding: 
where formula_82 denotes the sequence formula_83 with the term formula_84 omitted. 
When , and , the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of by , , , , , , the image of under the Plücker map is defined by the single equation
In general, however, many more equations are needed to define the Plücker embedding of a Grassmannian in projective space.
The Grassmannian as a real affine algebraic variety.
Let denote the Grassmannian of -dimensional subspaces of . Let denote the space of real matrices. Consider the set of matrices defined by if and only if the three conditions are satisfied:
 and are homeomorphic, with a correspondence established by sending to the column space of .
Duality.
Every -dimensional subspace of determines an -dimensional quotient space of . This gives the natural short exact sequence:
Taking the dual to each of these three spaces and linear transformations yields an inclusion of in with quotient :
Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence. Consequently there is a one-to-one correspondence between -dimensional subspaces of and -dimensional subspaces of . In terms of the Grassmannian, this is a canonical isomorphism
Choosing an isomorphism of with therefore determines a (non-canonical) isomorphism of and . An isomorphism of with is equivalent to a choice of an inner product, and with respect to the chosen inner product, this isomorphism of Grassmannians sends an -dimensional subspace into its -dimensional orthogonal complement.
Schubert cells.
The detailed study of the Grassmannians uses a decomposition into subsets called "Schubert cells", which were first applied in enumerative geometry. The Schubert cells for are defined in terms of an auxiliary flag: take subspaces , with . Then we consider the corresponding subset of , consisting of the having intersection with of dimension at least , for . The manipulation of Schubert cells is Schubert calculus.
Here is an example of the technique. Consider the problem of determining the Euler characteristic of the Grassmannian of -dimensional subspaces of . Fix a -dimensional subspace and consider the partition of into those -dimensional subspaces of that contain and those that do not. The former is and the latter is a -dimensional vector bundle over . This gives recursive formulas:
If one solves this recurrence relation, one gets the formula: if and only if is even and is odd. Otherwise:
Cohomology ring of the complex Grassmannian.
Every point in the complex Grassmannian manifold defines an -plane in -space. Fibering these planes over the Grassmannian one arrives at the vector bundle which generalizes the tautological bundle of a projective space. Similarly the -dimensional orthogonal complements of these planes yield an orthogonal vector bundle . The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of . In particular, all of the integral cohomology is at even degree as in the case of a projective space.
These generators are subject to a set of relations, which defines the ring. The defining relations are easy to express for a larger set of generators, which consists of the Chern classes of and . Then the relations merely state that the direct sum of the bundles and is trivial. Functoriality of the total Chern classes allows one to write this relation as
The quantum cohomology ring was calculated by Edward Witten in The Verlinde Algebra And The Cohomology Of The Grassmannian. The generators are identical to those of the classical cohomology ring, but the top relation is changed to
reflecting the existence in the corresponding quantum field theory of an instanton with fermionic zero-modes which violates the degree of the cohomology corresponding to a state by units.
Associated measure.
When is -dimensional Euclidean space, one may define a uniform measure on in the following way. Let be the unit Haar measure on the orthogonal group and fix in . Then for a set , define
This measure is invariant under actions from the group , that is, for all in . Since , we have . Moreover, is a Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.
Oriented Grassmannian.
This is the manifold consisting of all "oriented" -dimensional subspaces of . It is a double cover of and is denoted by:
As a homogeneous space it can be expressed as:
Applications.
Grassmann manifolds have found application in computer vision tasks of video-based face recognition and shape recognition. They are also used in the data-visualization technique known as the grand tour.
Grassmannians allow the scattering amplitudes of subatomic particles to be calculated via a positive Grassmannian construct called the amplituhedron.
The solution of the Kadomtsev–Petviashvili equations can be expressed as an infinite-dimensional Grassmann manifolds, where the KP equation is just a Plücker relation. Positive Grassmann manifolds can be used to achieve similar solutions of soliton solution of KP equation.