Grassmannian - The Grassmannian As A Homogeneous Space

The Grassmannian As A Homogeneous Space

The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space. First, recall that the general linear group GL(V) acts transitively on the r-dimensional subspaces of V. Therefore, if H is the set of stabilizers of this action, we have

Gr(r, V) = GL(V)/H.

If the underlying field is R or C and GL(V) is considered as a Lie group, then this construction makes the Grassmannian into a smooth manifold. It also becomes possible to use other groups to make this construction. To do this, fix an inner product on V. Over R, one replaces GL(V) by the orthogonal group O(V), and by restricting to orthonormal frames, one gets the identity

Gr(r, n) = O(n)/(O(r) × O(n–r)).

In particular, the dimension of the Grassmannian is r(n–r);.

Over C, one replaces GL(V) by the unitary group U(V). This shows that the Grassmannian is compact. These constructions also make the Grassmannian into a metric space: For a subspace W of V, let P_W be the projection of V onto W. Then

where denotes the operator norm, is a metric on Gr(r, V). The exact inner product used does not matter, because a different inner product will give an equivalent norm on V, and so give an equivalent metric.

If the ground field k is arbitrary and GL(V) is considered as an algebraic group, then this construction shows that the Grassmannian is a non-singular algebraic variety. It can be shown that H is a parabolic subgroup, from which it follows that Gr(r, V) is complete. It follows by the Veronese embedding that the Grassmannian is a projective variety, and more easily it follows from the Plücker embedding.