# Plücker embedding

In mathematics, the Plücker embedding is a method of realizing the Grassmannian ${\displaystyle \operatorname {Gr} _{k}(V)}$ of all k-dimensional subspaces of an n-dimensional vector space V as a subvariety of a projective space. More precisely, the Plücker map embeds ${\displaystyle \operatorname {Gr} _{k}(V)}$ algebraically into the projective space of the ${\displaystyle k}$th exterior power of that vector space, ${\displaystyle \mathbf {P} (\Lambda ^{k}V)}$. The image is the intersection of a number of quadrics defined by the Plücker relations.

The Plücker embedding was first defined in the case k = 2, n = 4 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). 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 ${\displaystyle \operatorname {Gr} _{k}(V)}$ under the Plücker embedding, relative to the natural basis in the exterior space ${\displaystyle \Lambda ^{k}V}$ corresponding to the natural basis in ${\displaystyle V=K^{n}}$ (where ${\displaystyle K}$ is the base field) are called Plücker coordinates.

## Definition

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

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

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.

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 relations called the Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of P(∧kV) and give another method of constructing the Grassmannian. To state the Plücker relations, let W be the k-dimensional subspace spanned by the basis of row vectors {w1, ..., wk}. Let ${\displaystyle W}$ be the ${\displaystyle k\times n}$ matrix of homogeneous coordinates whose rows are {w1, ..., wk} and let {W1, ..., Wn}, be the corresponding column vectors. For any ordered sequence ${\displaystyle 1\leq i_{1}<\cdots of ${\displaystyle k}$ positive integers, let ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ be the determinant of the ${\displaystyle k\times k}$ matrix with columns ${\displaystyle (W_{i_{1}},\dots ,W_{i_{k}})}$. Then ${\displaystyle \{W_{i_{1},\dots ,i_{k}}\}}$ are the Plücker coordinates of the element ${\displaystyle W}$ of the Grassmannian. They are the linear coordinates of the image ${\displaystyle \iota (W)}$ of ${\displaystyle W}$ under the Plücker map, relative to the standard basis in the exterior space ${\displaystyle \Lambda ^{k}V}$

For any two ordered sequences:

${\displaystyle i_{1}

of positive ${\displaystyle (k-1,k+1)}$ integers ${\displaystyle 1\leq i_{l},j_{m}\leq n}$, the following homogeneous equations are valid, and determine the image of W under the Plücker map:

${\displaystyle \sum _{l=1}^{k+1}(-1)^{l}W_{i_{1},\dots ,i_{k-1},j_{l}}W_{j_{1},\dots ,{\hat {j}}_{l},\dots j_{k+1}}=0,}$

where ${\displaystyle j_{1},\dots ,{\hat {j}}_{l}\dots j_{k+1}}$ denotes the sequence ${\displaystyle j_{1},\dots ,\dots j_{k+1}}$ with the term ${\displaystyle j_{l}}$ 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 P(∧kV) by W12, W13, W14, W23, W24, W34, the image of Gr(2, V) under the Plücker map is defined by the single equation

W12W34W13W24 + W14W23 = 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