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.

Related Topics:
Mathematics - Dimension - Subspace - Vector space - Hermann Grassmann - Projective space

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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).

Related Topics:
Perpendicular - Real projective plane

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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).

Related Topics:
Smooth manifold - Field - Topological space - Homogeneous space - Differential manifold - Algebraic variety - Up to - Isomorphism - Well-defined

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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.

Related Topics:
Real number - Complex number - Group action - GL(''V'') - Linear algebra - Block matrices - Topology

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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.

Related Topics:
Orthogonal group - Compact - Unitary group - Homotopy theory

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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.

Related Topics:
Algebraic group - Projective varieties - Homogeneous coordinates - Exterior power - Wedge product - Scalar - Identities - Minors

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