Transitivity on real Grassmannians

Statement

For $0\leq k\leq n$ , the natural action of $O(n)$ on the real Grassmannian $\operatorname{Gr}_k(\mathbb R^n)$ is transitive. For the coordinate plane $E=\mathbb R^k\oplus0$ , its stabilizer is $O(k)\times O(n-k)$ , so

\operatorname{Gr}_k(\mathbb R^n) \cong O(n)/(O(k)\times O(n-k)).

For $0<k<n$ , $SO(n)$ is also transitive, but its full stabilizer of the unoriented plane $E$ is

S(O(k)\times O(n-k)) =\{(A,B):\det(A)\det(B)=1\}.

Consequently,

\operatorname{Gr}_k(\mathbb R^n) \cong SO(n)/S(O(k)\times O(n-k)).

Proof mechanism

Given $k$ -planes $W,W'$ , choose orthonormal bases of them and of their orthogonal complements. The linear map carrying the first combined basis to the second is orthogonal and sends $W$ to $W'$ . If its determinant is $-1$ and $0<k<n$ , composing with a reflection that preserves $W'$ setwise changes the determinant without changing the image plane. This gives an element of $SO(n)$ .

An orthogonal transformation stabilizes $E$ exactly when it also stabilizes $E^{\perp}$ , hence is block diagonal with blocks in $O(k)$ and $O(n-k)$ . Intersecting this block group with $SO(n)$ imposes the product determinant condition above.

Connected versus full stabilizers

For $0<k<n$ , the identity component of the $SO(n)$ -stabilizer is

SO(k)\times SO(n-k),

where $SO(1)$ is the trivial group. The full stabilizer is generally larger: it also contains block pairs that reverse both the plane orientation and its complement orientation. Thus $SO(n)/(SO(k)\times SO(n-k))$ is not the ordinary Grassmannian; it is the oriented Grassmannian.

At $k=0$ or $k=n$ , the Grassmannian is a point and its stabilizer is all of the acting group. These boundary cases are excluded from the displayed $SO(n)$ stabilizer comparison.

References

Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, 2001. Publisher record.
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974, §§5–6. Publisher record.