Classification of topolgical phases when eigenstates belong to complex Grassmannian

You have filled all the negative energy one-particle states $|irangle$ and the many particle state you get is the wedge product $$ Psi= |1ranglewedge |2ranglewedge ldots wedge |nrangle $$ This state depends only on the subspace of $ {mathcal H}^{n+m}$ spanned by the states $|1rangle, |2rangle, ldots, |nrangle$ and this is left alone by the $U(n)times U(m)$ that transforms only the occupied space and the unoccupied space, but does not take any state from to the other. The set of all maps that takes one-Slater-determinant states to one-Slater determiant states is $U(n+m)$. The set of distinct one-Slater-determinant states is therefore the coset $U(n+m)/(U(n)times U(m))$.