Plücker embedding

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This article is about the general Plücker embedding. For the classical case of 2-planes in 4-space, see Plücker coordinates.

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 Rⁿ or Cⁿ, as a subvariety or submanifold of the projective space of the kth exterior power of that vector space, $\textstyle{\mathbf{P}(\bigwedge^k V)}$ .

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.

[edit] Definition

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

$\begin{align} \iota \colon \mathrm{Gr}_{k}(K^n) &{}\rightarrow \mathbb{P}(\wedge^k K^n)\\ \operatorname{span}( v_1, \ldots, v_k ) &{}\mapsto K( v_1 \wedge \cdots \wedge v_k ) \end{align}$

where Gr_k(Kⁿ) 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 (or Grassmann) coordinates that derives from linear algebra.