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In mathematics, the Plücker embedding describes a method to realize the Grassmannian of all ''r''-dimensional subspaces of an ''n''-dimensional vector space ''V'' as a subvariety of the projective space of the ''r''th exterior power of that vector space, P(∧''r'' ''V''). The Plücker embedding was first defined, in the case ''r'' = 2, ''n'' = 4, in coordinates by Julius Plücker as a way of describing the lines in three-dimensional space (which, as projective lines in real projective space, correspond to two-dimensional subspaces of a four-dimensional vector space). This was generalized by Hermann Grassmann to arbitrary ''r'' and ''n'' using a generalization of Plücker's coordinates, sometimes called Grassmann coordinates. == Definition == The Plücker embedding (over the field ''K'') is the map ''ι'' defined by : where Gr(''r'', ''K''''n'') is the Grassmannian, i.e., the space of all ''r''-dimensional subspaces of the ''n''-dimensional vector space, ''K''''n''. This is an isomorphism from the Grassmannian to the image of ''ι'', which is a projective variety. This variety can be completely characterized as an intersection of quadrics, each coming from a relation on the Plücker (or Grassmann) coordinates that derives from linear algebra. The bracket ring appears as the ring of polynomial functions on the exterior power. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Plücker embedding」の詳細全文を読む スポンサード リンク
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