Grassmannian
|
|
In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. The Grassmannian is named after Hermann Grassmann. The Grassmannian G1(V) is just the space of lines through the origin in V, that is, it is the projective space P(V). Grassmannians can therefore be thought of as generalizations of projective space.
When k = 2, the Grassmannian is the space of all planes through the origin. In Euclidean 3-space, a plane is completely characterized by the one and only line perpendicular to it (and vice-versa); hence G2,3 is isomorphic to G1,3 (both of which are isomorphic to the real projective plane).
Grassmannians often carry a natural geometrical structure derived from V. For example, when V is a real vector space the Grassmannian Gk,n can be given the structure of a smooth manifold of dimension k(n − k). For a fixed field K, we can consider for an n-dimensional vector space V, the set of subspaces with appropriate extra structure (e.g. a topological space, homogeneous space, differential manifold or algebraic variety), and notice that up to appropriate isomorphisms, we have a well-defined geometric object for the given pair (n,k).
Supposing first that K is the real number or complex number field, the easiest approach to Grassmannians is probably to consider them as homogeneous spaces. That is, the group action of GL(V) on the k-dimensional subspaces has a single orbit, as is shown in linear algebra. The stabilizer H of Kk in Kn, embedded using the first k co-ordinates, can be identified quickly as the block matrices defined by the condition aij = 0 for i = 1 to k and j > k (the upper right-hand block is 0). We can therefore identify Gk,n as the coset space GL(Kn)/H. This then provides a topology on the Grassmannian, and a smooth structure.
There can be other approaches: for example orthogonal groups also act transitively, so that the Grassmannians also appear as coset spaces for those groups. This shows directly that the real Grassmannians are compact (for the same result for complex Grassmannians one applies the unitary group). This representation might also be preferred in homotopy theory.
In the case of a general field K, something similar can be done with algebraic groups and their cosets. Then Grassmannians can be shown to be projective varieties. Explicit homogeneous coordinates are known, and come from the k-th exterior power: apply the wedge product to a basis of a k-dimensional subspace and the resulting k-vector is well-defined, up to a scalar multiple. It follows that the equations defining the Grassmannian can be regarded as the identities satisfied by k × k minors.
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 Gk,n are defined in terms of an auxiliary flag: take subspaces V1, V2, ..., Vk, with Vi contained in Vi+1. Then we consider the corresponding subset of Gk,n, consisting of the W having intersection with Vi of dimension at least i, for i = 1 to k.
See also
For an example of use of Grassmannians in differential geometry, see Gauss map and in projective geometry, see Plücker co-ordinates. Flag manifolds are generalizations of Grassmannians and Stiefel manifolds are closely related.