Plücker embedding

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In mathematics, the Plücker embedding is a method of realizing the Grassmannian of all k-dimensional subspaces of an n-dimensional vector space V as a subvariety of a projective space. More precisely, the Plücker map embeds algebraically into the projective space of the th exterior power of that vector space, . The image is the intersection of a number of quadrics defined by the Plücker relations.

The Plücker embedding was first defined in the case k = 2, n = 4 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). The image of that embedding is the Klein quadric in RP5.

Hermann Grassmann generalized Plücker's embedding to arbitrary k and n. The homogeneous coordinates of the image of the Grassmannian under the Plücker embedding, relative to the natural basis in the exterior space corresponding to the natural basis in (where is the base field) are called Plücker coordinates.

Definition[edit]

The Plücker embedding (over the field K) is the map ι defined by

where Gr(k, Kn) is the Grassmannian, i.e., the space of all k-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]

Plücker relations[edit]

The embedding of the Grassmannian satisfies some very simple quadratic relations called the Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of P(∧rV) and give another method of constructing the Grassmannian. To state the Plücker relations, let W be the r-dimensional subspace spanned by the basis of row vectors {w1, ..., wr}. Let be the matrix of homogeneous coordinates whose rows are {w1, ..., wr} and let {W1, ..., Wn}, be the corresponding column vectors. For any ordered sequence of positive integers, let be the determinant of the matrix with columns . Then are the Plücker coordinates of the element of the Grassmannian. They are the linear coordinates of the image of under the Plücker map, relative to the standard basis in the exterior space

For any two ordered sequences:

of positive integers , the following homogeneous equations are valid, and determine the image of W under the Plücker map:

where denotes the sequence with the term omitted.


When dim(V) = 4, and r = 2, the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of P(∧rV) by W12, W13, W14, W23, W24, W34, the image of Gr(2, V) under the Plücker map is defined by the single equation

W12W34W13W24 + W23W14 = 0.

In general, however, many more equations are needed to define the Plücker embedding of a Grassmannian in projective space.[2]

References[edit]

  1. ^ 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 
  2. ^ Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library (2nd ed.), New York: John Wiley & Sons, p. 211, ISBN 0-471-05059-8, MR 1288523, Zbl 0836.14001 

Further reading[edit]