```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.
```