grassmann manifold

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 $k(n-k).$

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 for the Grassmannian vary between authors; notations include , , , or to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space .

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 $(e_1, \dots, e_n)$, 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 $(W_1, \dots, W_k)$. The homogeneous coordinates of the element consist of the components of the rectangular matrix of maximal rank whose -th column vector is $W_i$, $i = 1, \dots, k$. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices and $\tilde$ represent the same element if and only if $\tilde = W g$ 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
$\begin 1 \\ & 1 \\ & & \ddots \\ & & & 1 \\ a_ & \cdots & \cdots & a_ \\ \vdots & & & \vdots \\ a_ & \cdots & \cdots & a_ \end.$
The matrix determines . In general, the first rows need not be independent, but for any whose rank is $k$, there exists an ordered set of integers $1 \le i_1 < \cdots < i_k \le n$ such that the submatrix $W_$ consisting of the $i_1, \ldots, i_k$-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 let $U_$ be the set of $n \times k$ matrices whose submatrix $W_$ is nonsingular, where the th row of $W_$ is the th row of . The coordinate function on $U_$ is then defined as the map $A^$ that sends to the rectangular matrix whose rows are the rows of the matrix $W W^_$ complementary to $(i_1, \dots, i_k)$. The choice of homogeneous coordinate matrix representing the element does not affect the values of the coordinate matrix $A^$ representing on the coordinate neighbourhood $U_$. Moreover, the coordinate matrices $A^$ may take arbitrary values, and they define a diffeomorphism from $U_$ onto the space of -valued matrices.

On the overlap
$U_ \cap U_$
of any two such coordinate neighborhoods, the coordinate matrix values are related by the transition relation
$A^ W_ = A^ W_,$
where both $W_$ and $W_$ are invertible. Hence the transition functions are differentiable, even a quotient of polynomials. Hence $(U_, A^)$ gives an atlas of as a differentiable, or even as an algebraic variety.

The Grassmannian as a set of orthogonal projections

An alternative way to define a real or complex Grassmannian as a real manifold is to consider it as an explicit set of orthogonal projections defined by explicit equations of full rank ( problem 5-C). For this choose a positive definite real or Hermitian inner product $\langle \cdot , \cdot \rangle$ on $V$ depending on whether is real or complex. A $k$-dimensional subspace $U$ now determines a unique orthogonal projection $P_U$ of rank $k$. Conversely, every projection $P$ of rank $k$ defines a subspace: its image $U_P = \mathrm(P)$. Since for a projection the rank equals its trace, we can define the Grassman manifold as an explicit set of projections
$\mathrm(k, V) = \left\$
In particular taking $V = \Reals^n$ or $V = \Complex^n$ this gives completely explicit equations for an embedding of the Grassmannian in the space of matrices $\Reals^$ respectively $\Complex^$.

As this defines the Grassmannian as a closed subset of the sphere $\$ this is one way to see that the Grassmannian is compact Hausdorff. This construction also makes the Grassmannian into a metric space: For a subspace of , let be the projection of onto . Then
$d(W, W') = \lVert P_W - P_ \rVert,$
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.

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 $\mathrm(V)$ acts transitively on the $r$-dimensional subspaces of $V$. Therefore, if $W \subset V$ is a subspace of $V$ of dimension $r$ and $H = \mathrm(W)$ is the stabilizer under this action, we have
:$\mathrm(r, V) = \mathrm(V)/H$
If the underlying field is $\mathbb$ or $\mathbb$ and $\mathrm(V)$ is considered as a Lie group, then this construction makes the Grassmannian into a smooth manifold. More generally, over a ground field $k$, the group $\mathrm(V)$ is an algebraic group, and 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, $H$ is a parabolic subgroup of $\mathrm(V)$.

Over $\mathbb$ or $\mathbb$ it also becomes possible to use other groups to make this construction. To do this over $\mathbb$, fix an inner product $q$ on $V$. The orthogonal group $O(V, q)$ acts transitively on the set of k-dimensional subspaces $\mathrm(k, V)$ and the stabiliser of a $k$-space $W$ is $O(W, q, _W)\times O(W^\perp, q, _)$. This gives the description as a homogeneous space
:$\mathrm(r, V) = O(V, q)/\left(O(W, q, _W)\times O(W^\perp q, _)\right)$.
If we take $V = \mathbb^n$ and $W = \mathbb^r \hookrightarrow \mathbb^n$ one gets the isomorphism
:$\mathrm(r,n) = O(n)/\left(O(r) \times O(n - r)\right)$

Over , one likewise chooses an Hermitian innerproduct $h$ and the unitary group $U(V, h)$ acts transitively, and one finds analogously
:$\mathrm(r, V) = U(V , h)/\left(U(W, h, _) \times U(W^\perp, _)\right)$
or for $V = \mathbb^n$ and $W = \mathbb^r \hookrightarrow \mathbb^n$
:$\mathrm(r, n) = U(n)/\left(U(r) \times U(n-r)\right)$

In particular, this again shows that the Grassmannian is a compact, and the (real or complex) dimension of the (real or complex) Grassmannian is .

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 $\mathcal E$ 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
$\mathcal E_T := \mathcal E \otimes_ O_T$
locally free of rank on . We denote this set by $\mathbf(r, \mathcal_T)$.

This functor is representable by a separated -scheme $\mathbf(r, \mathcal)$. The latter is projective if $\mathcal E$ is finitely generated. When is the spectrum of a field , then the sheaf $\mathcal E$ 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
$\mathbf(r, \mathcal E) \times_S S' \simeq \mathbf(r, \mathcal E_)$

In particular, for any point of , the canonical morphism , induces an isomorphism from the fiber $\mathbf(r, \mathcal E)_s$ to the usual Grassmannian $\mathbf(r, \mathcal E \otimes_ k(s))$ over the residue field .

Universal family

Since the Grassmannian scheme represents a functor, it comes with a universal object, $\mathcal G$, which is an object of
$\mathbf \left (r, \mathcal_ \right),$
and therefore a quotient module $\mathcal G$ of $\mathcal E_$, locally free of rank over $\mathbf(r, \mathcal)$. The quotient homomorphism induces a closed immersion from the projective bundle $\mathbf(\mathcal G)$:
$\mathbf(\mathcal G) \to \mathbf \left (\mathcal E_ \right) = \mathbf P() \times_S \mathbf(r, \mathcal E).$

For any morphism of -schemes:
$T \to \mathbf(r, \mathcal),$
this closed immersion induces a closed immersion
$\mathbf(\mathcal G_T) \to \mathbf (\mathcal) \times_S T.$

Conversely, any such closed immersion comes from a surjective homomorphism of -modules from $\mathcal E_T$ to a locally free module of rank . Therefore, the elements of $\mathbf(r, \mathcal E)(T)$ are exactly the projective subbundles of rank in
$\mathbf (\mathcal) \times_S T.$

Under this identification, when is the spectrum of a field and $\mathcal E$ is given by a vector space , the set of rational points $\mathbf(r, \mathcal)(k)$ correspond to the projective linear subspaces of dimension in , and the image of $\mathbf(\mathcal G)(k)$ in
$\mathbf(V) \times_k \mathbf(r, \mathcal E)$
is the set
$\left\.$

The Plücker embedding

The Plücker embedding is a natural embedding of the Grassmannian $\mathbf(k, V)$ into the projectivization of the exterior algebra :
$\iota : \mathbf(k, V) \to \mathbf \left(\Lambda^k V \right ).$

Suppose that is a -dimensional subspace of the -dimensional vector space . To define $\iota(W)$, choose a basis of , and let $\iota(W)$ be the wedge product of these basis elements:
\iota(W) = _1 \wedge \cdots \wedge w_k

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, $\iota$ is well-defined. To see that $\iota$ is an embedding, notice that it is possible to recover from $\iota$ as the span of the set of all vectors such that $w \wedge \iota (W) = 0$.

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 $\mathsf = \begin w_ &\cdots & w_\\ \vdots & \ddots & \vdots\\ w_ & \cdots & w_ \end,$ and let be the columns of $\mathsf$. For any ordered sequence $1\le i_1 < \cdots < i_k \le n$ of $k$ positive integers, let $W_$ be the determinant of the $k \times k$ matrix with columns $W_, \dots , W_$. The set $\$ is called the Plücker coordinates of the element $W$ of the Grassmannian (with respect to the basis of ). They are the linear coordinates of the image $\iota(W)$ of $W$ under the Plücker map, relative to the basis of the exterior power induced by the basis of .

For any two ordered sequences $1 \leq i_1 < i_2 \cdots < i_ \leq n$ and $1 \leq j_1 < j_2 \cdots < j_ \leq n$ of $k-1$ and $k+1$ positive integers, respectively, the following homogeneous equations are valid and determine the image of under the Plücker embedding:
$\sum_^ (-1)^\ell W_ W_ = 0,$
where $j_1, \ldots , \widehat, \ldots j_$ denotes the sequence $j_1, \ldots, j_$ with the term $j_\ell$ 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:

* is a projection operator: .
* is symmetric: .
* has trace : .

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:
$\chi_ = \chi_ + (-1)^r \chi_, \qquad \chi_ = \chi_ = 1.$

If one solves this recurrence relation, one gets the formula: if and only if is even and is odd. Otherwise:
$\chi_ = \binom.$

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
$c(E) c(F) = 1.$

The quantum cohomology ring was calculated by Edward Witten i
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
$c_k(E) c_(F) = (-1)^$
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
$\gamma_(A) = \theta_n\.$

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:
$\widetilde(r, n).$

As a homogeneous space it can be expressed as:
$\operatorname(n) / (\operatorname(r) \times \operatorname(n-r)).$

Applications

A key application of Grassmannians is as the "universal" embedding space
for bundles with connections on compact manifolds.

Solutions of the Kadomtsev–Petviashvili equation can be expressed in terms of abelian group flows on an infinite-dimensional Grassmann manifold. The KP equation, expressed in Hirota bilinear form in terms of the Tau function (integrable systems) is equivalent to the Plücker relations. Positive Grassmann manifolds can be used to express soliton solutions of KP equations which are nonsingular for real values of the KP flow parameters.

Grassmann manifolds have found applications 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 amplitude

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