I apologize in advance, as I do not have any formal background in representation theory, but the following spaces have come up in interesting problems, and I am trying to understand them. I am interested in the following two homogeneous spaces:
(1) $SU(2^n) / \otimes^n SU(2)$
(2) $U(2^n) / \otimes^n U(2)$
I believe that (1) has dimension (as a real manifold) $4^n -1 - 3^n$, and (2) has dimension $0$.
Question: Is there a standard way to compute what these spaces are?
In the case of (2), it should obviously be some discrete set of points but is it a single point, a finite set of points, or infinitely many points? In the case of (1), the situation is significantly more complex.
Even knowing the homotopy or homology of these spaces would be interesting. Ultimately, I am primarily interested in understanding the curvature of the spaces.
Any help is appreciated! Thanks!