Plücker embedding

In the mathematical fields of algebraic geometry and differential geometry (as well as representation theory), the Plücker embedding describes a method to realise the Grassmannian of all "k"-dimensional subspaces of a vector space "V", such as R^"n" or C^"n", as a subvariety or submanifold of the projective space of the "k"th exterior power of that vector space, .

The Plücker embedding was first defined, in the case "k" = 2, "n" = 4, in coordinates 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). This was generalized by Hermann Grassmann to arbitrary "k" and "n" using a generalization of Plücker's coordinates, sometimes called Grassmann coordinates.

Definition

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

:

where Gr_"k"(K^"n") is the Grassmannian, i.e., the space of all "k"-dimensional subspaces of the "n"-dimensional vector space, K^"n".

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.