Plücker coordinates

The coordinates of a straight line in three-dimensional space, the six numbers , and , of which the first three are the coordinates of the direction vector for the straight line and the second three are the moments of this vector about the origin. Let the line pass through the points and with projective coordinates and , respectively; the Plücker coordinates for this line are the numbers

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 , which are related to the Plücker ones as follows:

More generally, one naturally considers the Plücker coordinates as coordinates of a -dimensional vector subspace of an -dimensional vector space . Then they are understood as the set of numbers equal to -subdeterminants of the -matrix with as columns , , the coordinate columns (in some basis for ) of the basis vectors of a subspace . If are the components of a column , , then the Plücker coordinates (or Grassmann coordinates) are the numbers

The Plücker coordinates are anti-symmetric in all indices. The number of significant Plücker coordinates is .

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

A vector belongs to a subspace if the linear equations

with coefficients that are the Plücker coordinates for , are fulfilled. In these equations are all possible sets of numbers .

Comments

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

becomes

The Plücker coordinates of -dimensional subspaces of an -dimensional space (over any field) define an imbedding of the Grassmann variety into -dimensional projective space with . As a subvariety of , is given by quadratic relations, the Plücker relations, which look as follows:

i.e. take indices ; and write down the relation above, using that if two of the 's are equal. If , , there is just one relation: .

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