Parametrizations of degenerate density matrices

It turns out that a parametrization of degenerate density matrices requires a parametrization of $\mathfrak{F}=U(n)/({U(k_1)\times U(k_2)\times \cdots \times U(k_m)})\quad n=k_1 +\cdots + k_m$ where $U(k)$ denotes the set of all unitary $k\times k$-matrices with complex entries. Unfortunately the parametrization of this quotient space is quite involved. Our solution does not rely on Lie algebra methods {directly,} but succeeds through the construction of suitable sections for natural projections, by using techniques from the theory of homogeneous spaces. We mention the relation to the Lie algebra back ground and conclude with two concrete examples.