In mathematics, the Plücker map embeds the Grassmannian, whose elements are k-dimensional subspacess of an n-dimensional vector space V, either real or complex, in a projective space, thereby realizing it as a projective algebraic variety. More precisely, the Plücker map embeds-th exterior power. The image is algebraic, consisting of the intersection of a number of quadrics defined by the (see below).

The Plücker embedding was first defined by Julius Plücker in the case k=2, n=4 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 is the base field) are called Plücker coordinates.

Definition

, the Plücker embedding is the map ι defined by

is the projective equivalence class.

. The image can be completely characterized as the intersection of a number of quadrics, the Plücker quadrics (see below), which are expressed by homogeneous quadratic relations on the Plücker coordinates (see below) that derive from linear algebra.

The bracket ring.

Plücker relations

The image under the Plücker embedding satisfies a simple set of homogeneous quadratic relations, usually called the Plücker relations, or Grassmann–Plücker relations; this determinant is called a minor are the Plücker coordinates relative to the standard basis.

For any two ordered sequences:

under the Plücker map: is omitted. These are the Plücker relations.

When and by

under the Plücker map is defined by the single equation

In general, many more equations are needed to define the image of the Plücker embedding, as in (), but these are not, in general, algebraically independent

References

Further reading