Plücker embedding


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, by Julius Plücker as a way of describing the lines in three-dimensional 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 are called Plücker coordinates.

Definition

The Plücker embedding is the map ι defined by
where Gr 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 coordinates that derives from linear algebra.
The bracket ring appears as the ring of polynomial functions on the exterior power.

Plücker relations

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 and give another method of constructing the Grassmannian. To state the Plücker relations, let be the -dimensional subspace spanned by the basis of row vectors. Let be the matrix of
homogeneous coordinates whose rows are and let 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 under the Plücker map:
where denotes
the sequence with
the term omitted.
When, and, the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of by, the image of under the Plücker map is defined by the single equation
In general, however, many more equations are needed to define the Plücker embedding of a Grassmannian in projective space.