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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 rth 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.
The Plücker embedding (over the field K) is the map ι defined by
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.
- Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library (2nd ed.), New York: John Wiley & Sons, ISBN 0-471-05059-8, MR 1288523, Zbl 0836.14001