Plücker embedding
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.
Definition
The Plücker embedding (over the field K) is the map ι defined by
where Gr(r, Kn) is the Grassmannian, i.e., the space of all r-dimensional subspaces of the n-dimensional vector space, Kn.
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.[1]
References
- ↑ Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999), Oriented matroids, Encyclopedia of Mathematics and Its Applications 46 (2nd ed.), Cambridge University Press, p. 79, ISBN 0-521-77750-X, Zbl 0944.52006
- 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
Further reading
- Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics 227. New York, NY: Springer-Verlag. ISBN 0-387-23707-0. Zbl 1090.13001.