From: knighten@pinocchio (Bob Knighten) Newsgroups: sci.math Subject: Re: Plucker Coordinates Date: 26 Oct 89 23:46:13 GMT Reply-To: knighten@pinocchio (Bob Knighten) Organization: Encore Computer Corp
The set of all m-dimensional subspaces of an n-dimensional projective space naturally has the structure of Grassmann manifold. Plucker coordinates are coordinates for this manifold. The construction is to note that if a projective coordinate system is fixed for the n-dimensional projective space, then an m-dimensional subspace is determined by m+1 independent points x(0), . . ., x(m). Each of these points can be considered as an n+1-tuple indexed from 0 to n (the projective coordinates.) So for each set of m+1 distinct integers 0 <= i0 < i1 < . . . < im <= n one can form the determinant | x(0)(i0) x(0)(i1) . . . x(0)(im) | | x(1)(i0) x(1)(i1) . . . x(1)(im) | | . . . . . . | | . . . . . . | | x(m)(i0) x(m)(i1) . . . x(m)(im) | The tuple of all such determinants (with the canonical ordering) represents the subspace. This tuple provides homogeneous coordinates (i.e. two which differ only by a scalar multiple are identified) which are independent of the particular choice on independent points in the subspace. These coordinates are called Plucker (or Grassmann) coordinates. The individual components of the Plucker coordinates of a point in this Grassmann space are not independent - there is the Plucker relation which is essentially a restatement of expansion by minors. These were used extensively in the "classical" study of projective algebraic geometry. For example it is an immediate consequence of the existence of Plucker coordinates that the space of lines in projective 3-space can be identified with a quadric surface in projective 5-space. Strangely enough I can no longer remember where I learned this stuff and the only reference I know is C. Chevalley, Fundamental Concepts of Algebra Academic Press, 1956. pp. 201-203 which uses them to parametrize exterior algebras.