Grassmannian

In mathematics, the Grassmannian Gr(k, V) is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V. For example, the Grassmannian Gr(1, V) is the space of lines through the origin in V, so it is the same as the projective space of one dimension lower than V.[1][2]

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 Gr(2, R4) 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 Grk(V), Gr(k, V), Grk(n), or Gr(k, n) to denote the Grassmannian of k-dimensional subspaces of an n-dimensional vector space V.

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 M of dimension k embedded in Rn. At each point x in M, the tangent space to M can be considered as a subspace of the tangent space of Rn, which is just Rn. The map assigning to x its tangent space defines a map from M to Gr(k, n). (In order to do this, we have to translate the tangent space at each xM so that it passes through the origin rather than x, and hence defines a k-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 M, so that every vector bundle generates a continuous map from M 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.

For k = 1, the Grassmannian Gr(1, n) is the space of lines through the origin in n-space, so it is the same as the projective space of n − 1 dimensions.

For k = 2, 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 Gr(2, 3), Gr(1, 3), and P2 (the projective plane) may all be identified with each other.

Let V be an n-dimensional vector space over a field K. The Grassmannian Gr(k, V) is the set of all k-dimensional linear subspaces of V. The Grassmannian is also denoted Gr(k, n) or Grk(n).

Now we define a coordinate atlas. For any n × k matrix W, we can apply elementary column operations to obtain its reduced column echelon form. If the first k rows of W are linearly independent, the result will have the form

of any two such coordinate neighborhoods, the coordinate matrix values are related by the transition relation

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 GL(V) acts transitively on the r-dimensional subspaces of V. Therefore, if H is the stabilizer of any of the subspaces under this action, we have

If the underlying field is R or C and GL(V) 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 V. Over R, one replaces GL(V) by the orthogonal group O(V), and by restricting to orthonormal frames, one gets the identity

Over C, one replaces GL(V) by the unitary group U(V). This shows that the Grassmannian is compact. These constructions also make the Grassmannian into a metric space: For a subspace W of V, let PW be the projection of V onto W. Then

where ||⋅|| denotes the operator norm, is a metric on Gr(r, V). The exact inner product used does not matter, because a different inner product will give an equivalent norm on V, and so give an equivalent metric.

If the ground field k is arbitrary and GL(V) 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, H is a parabolic subgroup of GL(V).

In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.[4]

By construction, the Grassmannian scheme is compatible with base changes: for any S-scheme S′, we have a canonical isomorphism


When dim(V) = 4, and k = 2, the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of PkV) by W12, W13, W14, W23, W24, W34, the image of Gr(2, V) 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.[6]

Let Gr(r, Rn) denote the Grassmannian of r-dimensional subspaces of Rn. Let M(n, R) denote the space of real n × n matrices. Consider the set of matrices A(r, n) ⊂ M(n, R) defined by XA(r, n) if and only if the three conditions are satisfied:

A(r, n) and Gr(r, Rn) are homeomorphic, with a correspondence established by sending XA(r, n) to the column space of X.

Every r-dimensional subspace W of V determines an (nr)-dimensional quotient space V/W of V. This gives the natural short exact sequence:

Taking the dual to each of these three spaces and linear transformations yields an inclusion of (V/W) in V with quotient W:

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 r-dimensional subspaces of V and (nr)-dimensional subspaces of V. In terms of the Grassmannian, this is a canonical isomorphism

Choosing an isomorphism of V with V therefore determines a (non-canonical) isomorphism of Gr(r, V) and Gr(nr, V). An isomorphism of V with V is equivalent to a choice of an inner product, and with respect to the chosen inner product, this isomorphism of Grassmannians sends an r-dimensional subspace into its (nr)-dimensional orthogonal complement.

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 Gr(r, n) are defined in terms of an auxiliary flag: take subspaces V1, V2, ..., Vr, with ViVi + 1. Then we consider the corresponding subset of Gr(r, n), consisting of the W having intersection with Vi of dimension at least i, for i = 1, ..., r. 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 r-dimensional subspaces of Rn. Fix a 1-dimensional subspace RRn and consider the partition of Gr(r, n) into those r-dimensional subspaces of Rn that contain R and those that do not. The former is Gr(r − 1, n − 1) and the latter is a r-dimensional vector bundle over Gr(r, n − 1). This gives recursive formulas:

If one solves this recurrence relation, one gets the formula: χr,n = 0 if and only if n is even and r is odd. Otherwise:

Every point in the complex Grassmannian manifold Gr(r, n) defines an r-plane in n-space. Fibering these planes over the Grassmannian one arrives at the vector bundle E which generalizes the tautological bundle of a projective space. Similarly the (nr)-dimensional orthogonal complements of these planes yield an orthogonal vector bundle F. The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of E. 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 E and F. Then the relations merely state that the direct sum of the bundles E and F 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 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 2n fermionic zero-modes which violates the degree of the cohomology corresponding to a state by 2n units.

When V is n-dimensional Euclidean space, one may define a uniform measure on Gr(r, n) in the following way. Let θn be the unit Haar measure on the orthogonal group O(n) and fix W in Gr(r, n). Then for a set AGr(r, n), define

This measure is invariant under actions from the group O(n), that is, γr,n(gA) = γr,n(A) for all g in O(n). Since θn(O(n)) = 1, we have γr,n(Gr(r, n)) = 1. Moreover, γr,n 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.

This is the manifold consisting of all oriented r-dimensional subspaces of Rn. It is a double cover of Gr(r, n) and is denoted by:

Grassmann manifolds have found application in computer vision tasks of video-based face recognition and shape recognition.[7] 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.[8]

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.[9][10] Positive Grassmann manifolds can be used to achieve similar solutions of soliton solution of KP equation.[11][12]