Grassmannian
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In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an finite dimensional vector space V, often denoted Grk(V) or simply Grk(n) or G(k,n) when V is a standard n-dimensional vector space over a given field[1]. The Grassmannian is named after Hermann Grassmann. The Grassmannian Gr1(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.
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[edit] Motivation
By giving subspaces a topological structure it is possibly 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. Though such concepts may seem strangely out of place they can coincide with things that one is interested in, and can describe ideas that could not be considered otherwise - or at least describe them more economically.
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 
. At each point x in M, the tangent space to M can be considered as a subspace of the tangent space of , which is just . The map assigning to x its tangent space defines a map from M to Grk(n). (In order to do this, we have to translate the geometrical tangent space to M 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 with maps that are homotopic are isomorphic. But the definition of homotopic relies on a notion of continuity, and hence a topology.
These ideas are used in the definition of R-structures
[edit] Low dimensions
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 Gr2(3) is isomorphic to Gr1(3) (both of which are isomorphic to the real projective plane).
[edit] The structure carried by a Grassmannian
Grassmannians often carry a natural geometrical structure derived from V. For example, when V is a real vector space the Grassmannian Grk(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 Grk(n) as the coset space
- GL(Kn)/H.
This then provides a topological space structure on the Grassmannian, and, more than that, a smooth structure.
[edit] Other coset space representations
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.
[edit] Algebraic group approach
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, the Plücker or Grassmann coordinates, are known, and come from the kth 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, purely algebraically, as the identities satisfied by k × k minors.
[edit] 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 Grk(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 Grk(n), consisting of the W having intersection with Vi of dimension at least i, for i = 1 to k. The manipulation of Schubert cells is Schubert calculus.
[edit] 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.
[edit] References
- ^ In fact the range of possible notations is quite diverse: the Grassmannian can be denoted by G, Gr, or Gr, and the dimensions k and n can be subscripts, or can be bracketed (in either order).
- Joe Harris, Algebraic Geometry, A First Course, (1992) Springer, New York, ISBN 0-387-97716-3
