# Plücker coordinates

The coordinates of a straight line in three-dimensional space, the six numbers $p _ {01} , p _ {02} , p _ {03} , p _ {23} , p _ {31}$, and $p _ {12}$, of which the first three are the coordinates of the direction vector $l$ for the straight line $L$ and the second three are the moments of this vector about the origin. Let the line $L$ pass through the points $X$ and $Y$ with projective coordinates $( x _ {0} : \dots : x _ {3} )$ and $( y _ {0} : \dots : y _ {3} )$, respectively; the Plücker coordinates for this line are the numbers

$$p _ {ik} = x _ {i} y _ {k} - x _ {k} y _ {i} .$$

The Plücker coordinates are used in line geometry. They were first considered by J. Plücker (1869). Sometimes, instead of the Plücker coordinates one uses the Klein coordinates $( x _ {0} : \dots : x _ {5} )$, which are related to the Plücker ones as follows:

$$p _ {01} = x _ {0} + x _ {1} ,\ \ p _ {02} = x _ {2} + x _ {3} ,\ \ p _ {03} = x _ {4} + x _ {5} ,$$

$$p _ {23} = x _ {0} - x _ {1} ,\ p _ {31} = x _ {2} - x _ {3} ,\ p _ {12} = x _ {4} - x _ {5} .$$

More generally, one naturally considers the Plücker coordinates as coordinates of a $p$-dimensional vector subspace of an $n$-dimensional vector space $V$. Then they are understood as the set of numbers equal to $( p \times p)$-subdeterminants of the $( n \times p)$-matrix $A( a _ {1} \dots a _ {p} )$ with as columns $a _ {i}$, $1 \leq i \leq p$, the coordinate columns (in some basis for $V$) of the basis vectors of a subspace $W$. If $a _ {i} ^ {j}$ are the components of a column $a _ {i}$, $1 \leq i \leq p$, then the Plücker coordinates (or Grassmann coordinates) are the numbers

$$u ^ {i _ {1} \dots i _ {p} } = \left | \begin{array}{lll} a _ {1} ^ {i _ {i} } &\cdots &a _ {p} ^ {i _ {1} } \\ \vdots &\ddots &\vdots \\ a _ {1} ^ {i _ {p} } &\cdots &a _ {p} ^ {i _ {p} } \\ \end{array} \right | = \ p! a _ {1} ^ {[ i _ {1} } \dots a _ {p} ^ { {} i _ {p} ] } ,\ \ 1 \leq i _ \nu \leq n.$$

The Plücker coordinates are anti-symmetric in all indices. The number of significant Plücker coordinates is $( {} _ {p} ^ {n} )$.

When the basis of $W$ is changed and the basis for $V$ is fixed, the Plücker coordinates are all multiplied by the same non-zero number. When the basis of $V$ is changed and the basis for $W$ is fixed, the Plücker coordinates transform as the components of a contravariant tensor of valency $p$ (see Poly-vector). Two subspaces coincide if and only if their Plücker coordinates, calculated in the same basis for $V$, differ only by a non-zero factor.

A vector $x$ belongs to a subspace $W$ if the linear equations

$$\sum _ {\alpha = 1 } ^ { p+1 } (- 1) ^ {\alpha - 1 } x ^ {i _ \alpha } u ^ {i _ {1} \dots i _ {\alpha - 1 } i _ {\alpha + 1 } \dots i _ {p} } = 0,$$

with coefficients that are the Plücker coordinates for $W$, are fulfilled. In these equations $i _ {1} < \dots < i _ {p}$ are all possible sets of numbers $1 \dots n$.

Relating the Plücker and Klein coordinates as above, the Plücker identity

$$p _ {01} p _ {23} + p _ {02} p _ {31} + p _ {03} p _ {12} = 0$$

becomes

$$x _ {0} ^ {2} + x _ {2} ^ {2} + x _ {4} ^ {2} = \ x _ {1} ^ {2} + x _ {3} ^ {2} + x _ {5} ^ {2} .$$

The Plücker coordinates of $p$-dimensional subspaces $W$ of an $n$-dimensional space $V$ (over any field) define an imbedding of the Grassmann variety $G _ {p} ( V)$ into $N$-dimensional projective space $P ^ {N}$ with $N = ( {} _ {p} ^ {n} ) - 1$. As a subvariety of $P ^ {N}$, $G _ {p} ( V)$ is given by quadratic relations, the Plücker relations, which look as follows:

$$\sum _ { k=1 } ^ { p } (- 1) ^ {k} u ^ {i _ {1} \dots i _ {p-1} j _ {k} } u ^ {j _ {1} \dots \widehat{j _ {k} } \dots j _ {p+1} } = 0,$$

i.e. take $2p$ indices $1 \leq i _ {1} \dots i _ {p-1}$; $j _ {1} \dots j _ {p+1} \leq n$ and write down the relation above, using that $u ^ {k _ {1} \dots k _ {p} } = 0$ if two of the $k$'s are equal. If $p = 2$, $n = 4$, there is just one relation: $u ^ {12} u ^ {34} - u ^ {13} u ^ {24} + u ^ {14} u ^ {23} = 0$.

How to Cite This Entry:
Plücker coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pl%C3%BCcker_coordinates&oldid=51951
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article