Plücker Embedding

In mathematics, the Plücker map embeds the Grassmannian $\mathbf(k,V)$, 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 $\mathbf(k,V)$ into the projectivization
$\mathbf(\Lambda^k V)$ 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 $\mathbf(k,V)$ 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
$\mathbf(k, V)$ the Grassmannian of $k$-dimensional subspaces of $V$, the Plücker embedding is the map ''ι'' defined by

::\begin
\iota \colon \mathbf(k, V) &\rightarrow \mathbf(\Lambda^k V),\\
\iota \colon w:=\operatorname( w_1, \ldots, w_k ) &\mapsto w_1 \wedge \cdots \wedge w_k
\end

where $(w_1, \dots , w_k)$ is a basis for the element $w\in \mathbf(k, V)$ and w_1 \wedge \cdots \wedge w_k /math> 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 $\mathbf(\Lambda^k V)$. 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.

Plücker relations

The embedding of the Grassmannian satisfies some very simple quadratic relations usually called
the Plücker relations, or Grassmann–Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of $\mathbf(\Lambda^k V)$ and give another method of constructing the Grassmannian. To state the Grassmann–Plücker relations, let be the -dimensional subspace spanned by the basis of column vectors $W_1, \dots, W_k$.
Let $W$ be the $n \times k$ matrix of homogeneous coordinates,
whose columns are $W_1, \dots, W_k$. For any ordered sequence $1\le i_1 < \cdots < i_k \le n$
of $k$ integers, let $W_$ be the determinant of the
$k \times k$ matrix whose rows are the $i_1, \dots i_k$
rows of $W$. Then, up to projectivization, $\$
are the Plücker coordinates of the element /math> of the Grassmannian $\mathbf(k, V)$
whose homogeneous coordinates are $W$. They are the linear coordinates of the image $\iota(W)$ of
\in \mathbf(k, V) under the Plücker map, relative to the standard basis in the exterior space $\Lambda^k V$.

For any two ordered sequences:
::$i_1 < i_2 < \cdots < i_, \quad j_1 < j_2 < \cdots < j_$
of positive integers $1 \le i_l, j_m \le n$, the following homogeneous equations are valid, and determine the image of under the Plücker map:

where $j_1, \dots , \hat_l \dots j_$ denotes
the sequence $j_1, \dots , \dots j_$ with
the term $j_l$ omitted.

When and , the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of $\Lambda^2 V$ by
:: $W_ = -W_, \quad 1\le i,j, \le 4,$
the image of $\mathbf(2, V)$ under the Plücker map is defined by the single equation

::$W_W_ - W_W_ + W_W_=0.$

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

References

Further reading

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{{DEFAULTSORT:Plucker embedding
Algebraic geometry
Differential geometry