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.