Plücker embedding

In mathematics, the Plücker map embeds the Grassmannian ${\displaystyle \mathbf {Gr} (k,V)}$, whose elements are k-dimensional subspaces of an n-dimensional vector space V, in a projective space, thereby realizing it as an algebraic variety. More precisely, the Plücker map embeds ${\displaystyle \mathbf {Gr} (k,V)}$ into the projectivization ${\displaystyle \mathbf {P} (\Lambda ^{k}V)}$ of the ${\displaystyle k}$-th exterior power of ${\displaystyle V}$. 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 ${\displaystyle k=2,n=4}$ 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 ${\displaystyle \mathbf {Gr} (k,V)}$ under the Plücker embedding, relative to the 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

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

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

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

This is an embedding of the Grassmannian into the projectivization ${\displaystyle \mathbf {P} (\Lambda ^{k}V)}$. 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 the exterior power.[1]

Plücker relations

The embedding of the Grassmannian satisfies some very simple quadratic relations usually called the called Plücker relations, or Grassmann–Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of ${\displaystyle \mathbf {P} (\Lambda ^{k}V)}$ and give another method of constructing the Grassmannian. To state the Grassmann–Plücker relations, let W be the k-dimensional subspace spanned by the basis of column vectors ${\displaystyle W_{1},\dots ,W_{k}}$. Let ${\displaystyle W}$ be the ${\displaystyle n\times k}$ matrix of homogeneous coordinates, whose columns are ${\displaystyle W_{1},\dots ,W_{k}}$. For any ordered sequence ${\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}$ of ${\displaystyle k}$ integers, let ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ be the determinant of the ${\displaystyle k\times k}$ matrix whose rows are the ${\displaystyle i_{1},\dots i_{k}}$ rows of ${\displaystyle W}$. Then, up to projectivization, ${\displaystyle \{W_{i_{1},\dots ,i_{k}}\}}$ are the Plücker coordinates of the element ${\displaystyle [W]}$ of the Grassmannian ${\displaystyle \mathbf {Gr} (k,V)}$ whose homgeneous coordinates are ${\displaystyle W}$. They are the linear coordinates of the image ${\displaystyle \iota (W)}$ of ${\displaystyle [W]\in \mathbf {Gr} (k,V)}$ under the Plücker map, relative to the standard basis in the exterior space ${\displaystyle \Lambda ^{k}V}$.

For any two ordered sequences:

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

of positive 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:

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

(1)

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 ${\displaystyle \Lambda ^{2}V}$ by

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

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

${\displaystyle W_{12}W_{34}-W_{13}W_{24}+W_{14}W_{23}.}$

In general, many more equations, as in (1), are needed to define the image of the Plücker embedding[2] although these are not, in general, algebraically independent.

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,

Further reading

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