# Plücker embedding

In mathematics, the Plücker embedding is a method to realize the Grassmannian of all r-dimensional subspaces of an n-dimensional vector space V as a subvariety of the projective space of the rth exterior power of that vector space, P(∧r V).

The Plücker embedding was first defined, in the case r = 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). That embedding is to the Klein quadric in RP5

Hermann Grassmann generalized Plucker's embedding to arbitrary r and n; the coordinate generalization is sometimes called Grassmann coordinates.

## Definition

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

{\displaystyle {\begin{aligned}\iota \colon \mathbf {Gr} (r,K^{n})&{}\rightarrow \mathbf {P} (\wedge ^{r}K^{n})\\\operatorname {span} (v_{1},\ldots ,v_{r})&{}\mapsto K(v_{1}\wedge \cdots \wedge v_{r})\end{aligned}}}

where Gr(r, Kn) is the Grassmannian, i.e., the space of all r-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.

The bracket ring appears as the ring of polynomial functions on the exterior power.[1]

## Plücker relations

The embedding of the Grassmannian satisfies some very simple quadratic polynomials called the Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of P(∧rV) and give another method of constructing the Grassmannian. To state the Plücker relations, choose two r-dimensional subspaces W and Z of V with bases {w1, ..., wr}, and {z1, ..., zr}, respectively. Then, for any integer k ≥ 0, the following equation is true in the homogeneous coordinate ring of P(∧rV):

${\displaystyle \psi (W)\cdot \psi (Z)-\sum _{i_{1}<\cdots

When dim(V) = 4, and r = 2, the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of P(∧rV) by X1,2, X1,3, X1,4, X2,3, X2,4, X3,4, we have that Gr(2, V) is defined by the equation

X1,2X3,4X1,3X2,4 + X2,3X1,4 = 0.

In general, however, many more equations are needed to define the Plücker embedding of a Grassmannian in projective space.[2]

## References

1. ^ Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999), Oriented matroids, Encyclopedia of Mathematics and Its Applications, 46 (2nd ed.), Cambridge University Press, p. 79, ISBN 0-521-77750-X, Zbl 0944.52006
2. ^ Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library (2nd ed.), New York: John Wiley & Sons, p. 211, ISBN 0-471-05059-8, MR 1288523, Zbl 0836.14001