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 $.

Comments

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 $.

References

[a1]	H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 88–90
[a2]	B.L. van der Waerden, "Einführung in die algebraische Geometrie" , Springer (1939) pp. Chapt. 1