Information AboutGrassmannian |
| CATEGORIES ABOUT GRASSMANNIAN | |
| differential geometry | |
| projective geometry | |
| algebraic homogeneous spaces | |
| algebraic geometry | |
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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 ''G''2,3 is isomorphic to ''G''1,3 (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 ''G''''k'',''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 Isomorphism s, 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 ''K''''k'' in ''K''''n'', embedded using the first ''k'' co-ordinates, can be identified quickly as the Block Matrices defined by the condition ''a''''ij'' = 0 for ''i'' = 1 to ''k'' and ''j'' > ''k'' (the upper right-hand block is 0). We can therefore identify ''G''''k'',''n'' as the coset space GL(''K''''n'')/''H''. This then provides a Topology on the Grassmannian, and a smooth structure. There can be other approaches: for example Orthogonal Group s 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 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 . SCHUBERT CELLS The detailed study of the Grassmannians uses a decomposition into . SEE ALSO For an example of use of Grassmannians in Differential Geometry , see Gauss Map and in Projective Geometry , see Plücker Co-ordinates . Flag Manifold s are generalizations of Grassmannians and Stiefel Manifold s are closely related. |
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