In mathematics, the Plücker map embeds the Grassmannian
, whose elements are k-dimensional subspaces 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
The image is algebraic, consisting of the intersection of a number of quadrics defined by the § Plücker relations (see below).
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 basis in the exterior space
is the base field) are called Plücker coordinates.
, the Plücker embedding is the map ι defined by where
is the projective equivalence class of the element
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.
[1] The image under the Plücker embedding satisfies a simple set of homogeneous quadratic relations, usually called the Plücker relations, or Grassmann–Plücker relations, defining the intersection of a number of quadrics in
This shows that the Grassmannian embeds as an algebraic subvariety of
-dimensional subspace spanned by the basis represented by column vectors
matrix of homogeneous coordinates, whose columns are
of all such homogeneous coordinates matrices
related to each other by right multiplication by an invertible
are the Plücker coordinates of the element
They are the linear coordinates of the image
under the Plücker map, relative to the standard basis in the exterior space
Changing the basis defining the homogeneous coordinate matrix
just changes the Plücker coordinates by a nonzero scaling factor equal to the determinant of the change of basis matrix
, and hence just the representative of the projective equivalence class in
For any two ordered sequences: of positive integers
, the following homogeneous equations are valid, and determine the image of
These are generally referred to as the Plücker relations.
, the simplest Grassmannian which is not a projective space, and the above reduces to a single equation.
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 (1), but these are not, in general, algebraically independent.
The maximal number of algebraically independent relations (on Zariski open sets) is given by the difference of dimension between