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Grassmannian
المؤلف:
Fulton, W.
المصدر:
Schubert Varieties and Degeneracy Loci. New York: Springer-Verlag, 1998.
الجزء والصفحة:
...
7-7-2021
1845
Grassmannian
The Grassmannian is the set of
-dimensional subspaces in an
-dimensional vector space. For example, the set of lines
is projective space. The real Grassmannian (as well as the complex Grassmannian) are examples of manifolds. For example, the subspace
has a neighborhood
. A subspace
is in
if
and
and
. Then for any
, the vectors
and
are uniquely determined by requiring
and
. The other six entries provide coordinates for
.
In general, the Grassmannian can be given coordinates in a similar way at a point . Let
be the open set of
-dimensional subspaces which project onto
. First one picks an orthonormal basis
for
such that
span
. Using this basis, it is possible to take any
vectors and make a
matrix. Doing this for the basis of
, another
-dimensional subspace in
, gives a
-matrix, which is well-defined up to linear combinations of the rows. The final step is to row-reduce so that the first
block is the identity matrix. Then the last
block is uniquely determined by
. The entries in this block give coordinates for the open set
.
If is the standard basis of
, a basis of
is given by the
vectors
,
. If
is a basis of a subspace
of dimension
of
,
corresponds to a point
of
, whose coordinates are the components of
with respect to the basis of
given above. These coordinates are the so-called Grassmann coordinates of
.
A different choice of the basis of yields a different
-tuple of coordinates, which differs from the original
-tuple by a nonzero multiplicative constant, hence it corresponds to the same point.
The Grassmannian is also a homogeneous space. A subspace is determined by its basis vectors. The group that permutes basis vectors is . The matrix that fixes
is a diagonal block matrix, with a
nonsingular matrix in the top left, and a
invertible matrix in the lower right.
acts transitively on the Grassmannian
. Let
be the stabilizer (or isotropy) of
. Then
is the subgroup of
consisting of matrices
such that
for all
,
such that
and
.
is isomorphic to
.
The tangent space to the Grassmannian is given by matrices, i.e., linear maps from
to the quotient vector space
.
The elements are the
-minors of the
matrix whose
th row contains the components of
with respect to the basis
. It corresponds to a linear transformation
whose range is
. In general, the range of such a linear transformation has dimension
iff the corresponding
matrix has rank
.
Let be the subset of
defined by the condition that all the
-minors of the matrix
(which can be viewed as a sequence of
coordinates) be equal to zero, and one
-minor be nonzero. The Grassmannian
can be viewed as the image of the map
which maps each matrix of
to the sequence of its
-minors.
It as an algebraic projective algebraic variety defined by equations called Plücker's equations. It is a nonsingular variety of dimension .
REFERENCES:
Fulton, W. Schubert Varieties and Degeneracy Loci. New York: Springer-Verlag, 1998.
Harris, J. "Grassmannians and Related Varieties." Lecture 6 in Algebraic Geometry: A First Course. New York: Springer-Verlag, pp. 63-71, 1992.
Kleiman, S. and Laksov, D. "Schubert Calculus." Amer. Math. Monthly 79, 1061-1082, 1972.
Shafarevich, I. R. Basic Algebraic Geometry, Vol. 1, 2nd ed. Berlin: Springer-Verlag, pp. 42-44, 1994.
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