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Shapiro's Conjecture
Plücker Coordinates for the Grassmannian
There are at least two coordinate schemes for the set of lines in 3-space,
both of which generalize to other Grassmannians, and to the flag manifolds
which are the subject of this web page.
First, are the global Plücker coordinates.
It is easiest to work not with lines in projective 3-space
P3, but with 2-dimensional linear subspaces
of the 4-dimensional vector space
K4, where K is our field.
Such a 2-plane H is the row space of a 2 by 4 matrix, called a
representation matrix for H and also written
H.
Let the entries be
xi,j for
i=1,2 and j=0,1,2,3.
It is a classic fact (see Chapter 2
of [PW]) that the linear subspace
H is determined by the sextuple of the 2 by 2 minors of the matrix
H, up to a scalar.
This sextuple is called the Plücker coordinates for the 2-plane H.
They are
pi,j
:=
x1,i
x2,j
-
x1,j
x2,i .
The association of a 2-plane H to its Plücker coordinates
gives an embedding of the Grassmannian of lines in projective 3-space
(equivalently, of 2-dimensional linear subspaces of 4-space) into a
5-dimensional projective space.
The image is exactly those sextuples
[p01, p02, p03, p12, p13, p23]
which satisfy the quadratic Plücker equation
p03
p12
- p13
p02
+ p01
p23
= 0 .
Another way to give coordinates assumes that the first Plücker coordinate
p01 does not vanish.
Then dividing the Plücker coordinates by
p01 gives a vector
(1, p02, p03, p12, p13, p23),
where
p23 = p13p02 - p03p12.
Thus the four coordinates
(p02, p03, p12, p13)
determine H.
Another way to see this is that such 2-planes H are those
with a representation matrix of the form
[I2 : Y],
where I2 is the 2 by 2 identity matrix and
Y is the 2 by 2 matrix whose entries are
-p12,
-p13 in its first row and
p02,
p03 in its second row.
We call Y the standard local coordinates for the Grassmannian.
Last Modified Saturday 20 September 2003
by Frank Sottile