In mathematics, the Plücker map embeds the Grassmannian , whose elements are k-dimensional subspaces of an n-dimensional vector space V, in a projective space, thereby realizing it as an algebraic variety. More precisely, the Plücker map embeds into the projectivization of the -th exterior power of . The image is algebraic, consisting of the intersection of a number of quadrics defined by the Plücker relations (see below).
The Plücker embedding was first defined by Julius Plücker in the case 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 corresponding to the natural basis in (where is the base field) are called Plücker coordinates.
Denoting by the -dimensional vector space over the field , and by the Grassmannian of -dimensional subspaces of , the Plücker embedding is the map ι defined by
where is a basis for the element and is the projective equivalence class of the element of the th exterior power of .
This is an embedding of the Grassmannian into the projectivization . 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 appears as the ring of polynomial functions on the exterior power.
The embedding of the Grassmannian satisfies some very simple quadratic relations usually called the Plücker relations, or Grassmann–Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of and give another method of constructing the Grassmannian. To state the Grassmann–Plücker relations, let W be the k-dimensional subspace spanned by the basis of column vectors . Let be the matrix of homogeneous coordinates, whose columns are . For any ordered sequence of integers, let be the determinant of the matrix whose rows are the rows of . Then, up to projectivization, are the Plücker coordinates of the element of the Grassmannian whose homogeneous coordinates are . 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 k = 2, 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, many more equations, as in (1), are needed to define the image of the Plücker embedding although these are not, in general, algebraically independent.
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