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Plücker coordinates

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In mathematics Plücker co-ordinates are a way to assign to each line in projective 3-space a point in projective 5-space. Plücker co-ordinates were introduced by Julius Plücker in 1844.

Writing each point in homogeneous co-ordinates the Plücker Mapping is defined as follows:

$\begin{matrix} \phi : \mathbb{P}^3 \times \mathbb{P}^3 &\rightarrow& \mathbb{P}^5 \\ (x_0:x_1:x_2:x_3) \times (y_0:y_1:y_2:y_3) &\mapsto& (p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23}) \end{matrix}$

where $p i j = x i y j - x j y i$

This can be expressed in a more compact form. Consider the matrix:

$\begin{pmatrix} x_0 & x_1 & x_2 & x_3 \\ y_0 & y_1 & y_2 & y_3 \end{pmatrix}$

Then $p i j$ is just seen to be the determinant of the 2x2 matrix consisting of the i^th and j^th columns.

Contents

1 Properties of the Plücker mapping

1.1 The Plücker mapping is well defined
1.2 The Plücker mapping takes lines to points
1.3 The image of the Plücker map

2 Uses of the Plücker map

1 Line geometry

1.1 Other formulations of Plücker co-ordinates

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Properties of the Plücker mapping

For the following arguments, $X = (x_0:x_1:x_2:x_3) \in \mathbb{P}^3$ , $Y = (y_0:y_1:y_2:y_3) \in \mathbb{P}^3$ and $\phi(X,Y) = (p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23}) \in \mathbb{P}^5$ as defined above.

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The Plücker mapping is well defined

This is only true for $X \ne Y$ .

First, we notice that not all the p_ij are zero ( (0:0:0:0:0) is not a point of $\mathbb{P}^5$ ). This is true because if $y_{k_1}$ is a nonzero component of y, then there exists k₂ with $k_2 \ne k_1$ with $x_{k_2} \ne \frac{x_{k_1}}{y_{k_1}}y_{k_2}$ (otherwise $X = \frac{x_{k_1}}{y_{k_1}}Y = Y \in \mathbb{P}^3$ ).

Then the component of $φ(X, Y)$ , $p_{k_1k_2} = x_{k_1}y_{k_2} - x_{k_2}y_{k_1} = y_{k_1}( \frac{x_{k_1}}{y_{k_1}}y_{k_2} - x_{k_2} ) \ne 0$ .

Next, if $X = X^\prime$ and $Y = Y^\prime \in \mathbb{P}^3$ then $X^\prime =\lambda X$ and $Y^\prime = \mu Y$ .

So $\phi(X^\prime,Y^\prime) = (p_{01}^\prime:p_{02}^\prime:p_{03}^\prime:p_{12}^\prime:p_{13}^\prime:p_{23}^\prime)$

where $p_{ij}^\prime = x_i^\prime y_j^\prime - x_j^\prime y_i^\prime = \lambda x_i \mu y_j - \lambda x_j \mu y_i = \lambda \mu( x_i y_j - x_j y_i ) = \lambda \mu p_{ij}$ so

$\phi(X^\prime,Y^\prime) = \lambda \mu (p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23}) = (p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23}) = \phi( X, Y ) \in \mathbb{P}^5$ .

So $φ$ is well defined if $X \ne Y \in \mathbb{P}^3$ .

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The Plücker mapping takes lines to points

If $X^\prime$ and $Y^\prime$ are on the line determined by X and Y in $\mathbb{P}^3$ then

$X^\prime = ( x_0 + ry_0: x_1 + ry_1: x_2 + ry_2: x_3 + ry_3 )$ and

$Y^\prime = ( x_0 + sy_0: x_1 + sy_1: x_2 + sy_2: x_3 + sy_3 )$ for some r,s with $r \ne s$ .

Then the ij^th component of $\phi( X^\prime, Y^\prime )$ is $(x i + r y i)(x j + r y j) - (x j + r y j)(x i + s y i)$ . Expanding, this product, we have

(x i + r y i)(x j + r y j) - (x j + r y j)(x i + s y i) = (s - r)(x i y j - y j x i) = (s - r) p i j

So $\phi( X^\prime, Y^\prime ) = (s-r) \phi( X,Y ) = \phi( X,Y ) \in \mathbb{P}^5$

So $φ$ takes points on the same line in $\mathbb{P}^3$ to the same point in $\mathbb{P}^5$ .

For the other direction, we wish to show that if $\phi( X,Y ) = \phi( X^\prime, Y^\prime ) = ( p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23} )$ then $(X, Y)$ and $(X^\prime,Y^\prime)$ determine the same line in $\mathbb{P}^3$ .

This follows easily from a simple observation:

$\begin{matrix} y_0X - x_0Y & = & ( 0: x_1y_0 - x_0y_1: x_2y_0 - x_0y_2 : x_3y_0 - y_3x_0 ) = ( 0: -p_{01} : -p_{02} : -p_{03} ) \\ y_0^\prime X^\prime - x_0^\prime Y^\prime & = & ( 0: x_1^\prime y_0^\prime x_0^\prime y_1^\prime : x_2^\prime y_0^\prime - x_0^\prime y_2^\prime : x_3^\prime y_0^\prime - x_0^\prime y_3^\prime ) = ( 0: -p_{01} : -p_{02} : -p_{03} ) \\ \end{matrix}$

Similar calculations yield that $y_iX -x_iY = y_i^\prime X^\prime - y_i^\prime Y^\prime$

But $y i X - x i Y$ is on the line defined by $X, Y$ and $y_i^\prime X^\prime - x_i^\prime Y^\prime$ are on the line defined by $X^\prime,Y^\prime$ . Since the lines have more than two points in common, they must be the same.

So we can think of $φ(X, Y)$ as the line (in $\mathbb{P}^3$ ) through X and Y.

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The image of the Plücker map

The image of the Plücker map in $\mathbb{P}^5$ is called the Plücker quadric.

Define $C \subset \mathbb{P}^5$ by $C :{(p 01 : p 02 : p 03 : p 12 : p 13 : p 23) | p 01 p 23 - p 02 p 13 + p 03 p 12 = 0}$

A direct calculation yields that the image of a line $(p 01 : p 02 : p 03 : p 12 : p 23)$ satisfies the equation $p 01 p 23 - p 02 p 13 + p 03 p 12 = 0$

So the image of $φ$ is contained in C.

For a point $(p 01 : p 02 : p 03 : p 12 : p 13 : p 23)$ in C, one of the components must be nonzero. So we suppose $p_{01} \ne 0$ otherwise we can change coordinates.

Now let $X = (0: p 01 : p 02 : p 03)$ and $Y = (p 01 :0: - p 12 : - p 13)$

Then $\phi( X,Y ) = (-p_{01}^2:-p_{01}p_{02}:-p_{01}p_{03}:-p_{01}p_{12}:-p_{01}p_{13}:-p_{02}p_{13}+p_{12}p_{03} )$ using our relation $p 01 p 23 - p 02 p 13 + p 03 p 12 = 0$ we have $- p 02 p 13 + p 12 p 03 = - p 01 p 23$ so $\phi( X, Y ) = (-p_{01}^2:-p_{01}p_{02}:-p_{01}p_{03}:-p_{01}p_{12}:-p_{01}p_{13}:-p_{01}p_{23} ) = (p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23})$

So the Plücker map maps onto C.

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Uses of the Plücker map

Using the Plücker Map we can think of certain points in $\mathbb{P}^5$ as lines in $\mathbb{P}^3$ . We can use this characterization to easily find certain classes of lines.

If we want all the lines through the point $(x_0:x_1:x_2:x_3) \in \mathbb{P}^3$ , a direct calculation yields that these are just the lines with Plucker co-ordinates $(p 01 : p 02 : p 03 : p 12 : p 13 : 23)$ where $x 0 x 2 p 13 - x 0 x 1 p 23 - x 1 x 3 p 02 = 0$

If we want all the lines in the plane $x 0 = a 1 x 1 + a 2 x 2 + a 3 x 3$ , another calculation shows that these are just the points with Plucker co-ordinates satisfying $a 1 p 01 + a 2 p 02 + a 3 p 03 = 0$

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Line geometry

During the nineteenth century, line geometry was studied quite intensively. In terms of the formulation given above, this is a description of the intrinsic five-dimensional geometry on the quadric.

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Other formulations of Plücker co-ordinates

Plucker co-ordinates can be formulated more generally using the ideas of multilinear algebra and the wedge product. This leads to a simple formulation of the generalisation of the Plücker mapping.

A Plücker map is a map $π$

$\begin{matrix} \pi : Gr(n,N) &\rightarrow& \mathbb{P}(\wedge^n\mathbb{C}^N)\\ span( v_1, \ldots, v_n ) &\mapsto& \mathbb{C}( v_1 \wedge \ldots \wedge v_n )\\ \end{matrix}$

where Gr(n,N) is the Grassmannian, i.e. the space of all n-dimensional subspaces of an N-dimensional vector space. The map described above is the particular case where n = 1, N = 3.

In the general context, the Grassmannian can be completely characterised as an intersection of quadrics, each coming from a relation on the Plücker co-ordinates that derives from linear algebra.

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