(named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all
is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension
Giving them the further structure of a differentiable manifold, one can talk about smooth choices of subspace.
A natural example comes from tangent bundles of smooth manifolds embedded in a Euclidean space.
In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic.
We may apply column operations to reduce this submatrix to the identity matrix, and the remaining entries uniquely determine
An alternative way to define a real or complex Grassmannian as a manifold is to view it as a set of orthogonal projection operators (Milnor & Stasheff (1974) problem 5-C).
Since the rank of an orthogonal projection operator equals its trace, we can identify the Grassmann manifold
For the case of real or complex Grassmannians, the following is an equivalent way to express the above construction in terms of matrices.
is The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space.
is considered as a Lie group, this construction makes the Grassmannian a smooth manifold under the quotient structure.
In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.
, the Grassmannian functor associates the set of quotient modules of locally free of rank
-component column vectors forming the transpose of the corresponding homogeneous coordinate matrix: For any ordered sequence
, this single Plücker relation is In general, many more equations are needed to define the image
The detailed study of Grassmannians makes use of a decomposition into affine subpaces called Schubert cells, which were first applied in enumerative geometry.
have the following dimensions These are affine spaces, and their closures (within the Zariski topology) are known as Schubert varieties.
The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of
The defining relations are easy to express for a larger set of generators, which consists of the Chern classes of
Functoriality of the total Chern classes allows one to write this relation as The quantum cohomology ring was calculated by Edward Witten.
[7] 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
As a homogeneous space it can be expressed as: Given a real or complex nondegenerate symmetric bilinear form
for which Maximal isotropic Grassmannians with respect to a real or complex scalar product are closely related to Cartan's theory of spinors.
[8] Under the Cartan embedding, their connected components are equivariantly diffeomorphic to the projectivized minimal spinor orbit, under the spin representation, the so-called projective pure spinor variety which, similarly to the image of the Plücker map embedding, is cut out as the intersection of a number of quadrics, the Cartan quadrics.
[8][9][10] A key application of Grassmannians is as the "universal" embedding space for bundles with connections on compact manifolds.
[11][12] Another important application is Schubert calculus, which is the enumerative geometry involved in calculating the number of points, lines, planes, etc.
Subvarieties of Schubert cells can also be used to parametrize simultaneous eigenvectors of complete sets of commuting operators in quantum integrable spin systems, such as the Gaudin model, using the Bethe ansatz method.
These can be expressed in terms of abelian group flows on an infinite-dimensional Grassmann manifold.
[18][17] A similar construction holds for solutions of the BKP integrable hierarchy, in terms of abelian group flows on an infinite dimensional maximal isotropic Grassmann manifold.
[20][21][22] The scattering amplitudes of subatomic particles in maximally supersymmetric super Yang-Mills theory may be calculated in the planar limit via a positive Grassmannian construct called the amplituhedron.