The unitary group, U(n)$U(n)$, acts transitively on the Grassmann manifold X = Gr(2, C^n)$X = Gr(2, C^n)$. The isotropy group is H = U(2)xU(n-2) =$H = U(2)\times U(n-2)$, i.e. the group elements leaving some x$x$ fixed. What are the dimensions of the orbits of H$H$ in X$X$? The brute force calculation gets messy for some of the orbits, so I am wondering if these results are published somewhere. The generic orbit is codimension 2, and there are the special orbits of {x}$\{x\}$ and x_perp = Gr(2,C^{n-2})$x_\perp = Gr(2,C^{n-2})$, but there are other orbits, too.
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