Plücker embedding

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See also: Plücker coordinates and Grassmannian

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, such as Rⁿ or Cⁿ, as a subvariety or submanifold of the projective space of the kth exterior power of that vector space. It thus realises the Grassmannian in Rⁿ or Cⁿ as a projective variety in $P(\wedge^k\mathbb R^n)$ or $P(\wedge^k\mathbb C^n)$ .

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 complex numbers) is the map ι defined by

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

where Gr_k(Cⁿ) is the Grassmannian, i.e., the space of all k-dimensional subspaces of the n-dimensional complex vector space, Cⁿ.

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.