Plücker embedding

In mathematics, the Plücker map embeds the Grassmannian $\mathbf {Gr} (k,V)$, whose elements are k-dimensional subspaces of an n-dimensional vector space V, either real or complex, in a projective space, thereby realizing it as a projective algebraic variety. More precisely, the Plücker map embeds $\mathbf {Gr} (k,V)$ into the projectivization $\mathbf {P} (\Lambda ^{k}V)$ of the $k$-th exterior power of $V$. The image is algebraic, consisting of the intersection of a number of quadrics defined by the § Plücker relations (see below).

The Plücker embedding was first defined by Julius Plücker in the case $k=2,n=4$ 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 $\mathbf {Gr} (k,V)$ under the Plücker embedding, relative to the 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

Denoting by $V=K^{n}$ the $n$-dimensional vector space over the field $K$, and by $\mathbf {Gr} (k,V)$ the Grassmannian of $k$-dimensional subspaces of $V$, the Plücker embedding is the map ι defined by

${\begin{aligned}\iota \colon \mathbf {Gr} (k,V)&{}\rightarrow \mathbf {P} (\Lambda ^{k}V),\\\iota \colon {\mathcal {W}}:=\operatorname {span} (w_{1},\ldots ,w_{k})&{}\mapsto [w_{1}\wedge \cdots \wedge w_{k}],\end{aligned}}$

where $(w_{1},\dots ,w_{k})$ is a basis for the element ${\mathcal {W}}\in \mathbf {Gr} (k,V)$ and $[w_{1}\wedge \cdots \wedge w_{k}]$ is the projective equivalence class of the element $w_{1}\wedge \cdots \wedge w_{k}\in \Lambda ^{k}V$ of the $k$th exterior power of $V$.

This is an embedding of the Grassmannian into the projectivization $\mathbf {P} (\Lambda ^{k}V)$. The image can be completely characterized as the intersection of a number of quadrics, the Plücker quadrics (see below), which are expressed by homogeneous quadratic relations on the Plücker coordinates (see below) that derive from linear algebra.

The bracket ring appears as the ring of polynomial functions on $\Lambda ^{k}V$.

Plücker relations

The image under the Plücker embedding satisfies a simple set of homogeneous quadratic relations, usually called the Plücker relations, or Grassmann–Plücker relations, defining the intersection of a number of quadrics in $\mathbf {P} (\Lambda ^{k}V)$. This shows that the Grassmannian embeds as an algebraic subvariety of $\mathbf {P} (\Lambda ^{k}V)$ and gives another method of constructing the Grassmannian. To state the Grassmann–Plücker relations, let ${\mathcal {W}}\in \mathbf {Gr} (k,V)$ be the $k$-dimensional subspace spanned by the basis represented by column vectors $W_{1},\dots ,W_{k}$. Let $W$ be the $n\times k$ matrix of homogeneous coordinates, whose columns are $W_{1},\dots ,W_{k}$. Then the equivalence class $[W]$ of all such homogeneous coordinates matrices $Wg\sim W$ related to each other by right multiplication by an invertible $k\times k$ matrix $g\in \mathbf {GL} (k,K)$ may be identified with the element ${\mathcal {W}}$. For any ordered sequence $1\leq i_{1}<\cdots <i_{k}\leq n$ of $k$ integers, let $W_{i_{1},\dots ,i_{k}}$ be the determinant of the $k\times k$ matrix whose rows are the rows $(i_{1},\dots i_{k})$ of $W$. Then, up to projectivization, $\{W_{i_{1},\dots ,i_{k}}\}$ are the Plücker coordinates of the element ${\mathcal {W}}\sim [W]\in \mathbf {Gr} (k,V)$ whose homogeneous coordinates are $W$. They are the linear coordinates of the image $\iota ({\mathcal {W}})$ of ${\mathcal {W}}\in \mathbf {Gr} (k,V)$ under the Plücker map, relative to the standard basis in the exterior space $\Lambda ^{k}V$. Changing the basis defining the homogeneous coordinate matrix $M$ just changes the Plücker coordinates by a nonzero scaling factor equal to the determinant of the change of basis matrix $g$, and hence just the representative of the projective equivalence class in $\Lambda ^{k}V$.

For any two ordered sequences:

$i_{1}<i_{2}<\cdots <i_{k-1},\quad j_{1}<j_{2}<\cdots <j_{k+1}$

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

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

(1)

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. These are generally referred to as the Plücker relations.

When dim(V) = 4 and k = 2, we get $\mathbf {Gr} (2,V)$, the simplest Grassmannian which is not a projective space, and the above reduces to a single equation. Denoting the coordinates of $\Lambda ^{2}V$ by

$W_{ij}=-W_{ji},\quad 1\leq i,j\leq 4,$

the image of $\mathbf {Gr} (2,V)$ under the Plücker map is defined by the single equation

$W_{12}W_{34}-W_{13}W_{24}+W_{14}W_{23}=0.$

In general, many more equations are needed to define the image of the Plücker embedding, as in (1), but these are not, in general, algebraically independent. The maximal number of algebraically independent relations (on Zariski open sets) is given by the difference of dimension between $\mathbf {P} (\Lambda ^{k}V)$ and $\mathbf {Gr} (k,V)$, which is ${\tbinom {n}{k}}-k(n-k)-1.$

Further reading

• Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. Vol. 227. New York, NY: Springer-Verlag. ISBN 0-387-23707-0. Zbl 1090.13001.