# Plücker embedding

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In mathematics, the Plücker embedding is a method of realizing the Grassmannian $Gr_{k}(V)$ 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 $Gr_{k}(V)$ algebraically into the projective space of the $k$ th exterior power of that vector space, $P(\Lambda ^{k}V)$ . 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 $Gr_{k}(V)$ under the Plücker embedding, relative to the natural basis in the exterior space $\Lambda ^{k}V$ corresponding to the natural basis in $V=K^{n}$ (where $K$ is the base field) are called Plücker coordinates.

## Definition

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

{\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.

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

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

$\sum _{l=1}^{k+1}(-1)^{l}W_{i_{1},\dots ,i_{k-1},j_{l}}W_{j_{1},\dots ,{\hat {j}}_{l},\dots j_{k+1}}=0,$ where $j_{1},\dots ,{\hat {j}}_{l}\dots j_{k+1}$ denotes the sequence $j_{1},\dots ,\dots j_{k+1}$ with the term $j_{l}$ omitted.

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 W12, W13, W14, W23, W24, W34, the image of Gr(2, V) under the Plücker map is defined by the single equation

W12W34W13W24 + W23W14 = 0.

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