  We give a one-to-one correspondence between classes of density matrices under
local unitary invariance and the double cosets of unitary groups. We show that
the interrelationship among classes of local unitary equivalent multi-partite
mixed states is independent from the actual values of the eigenvalues and only
depends on the multiplicities of the eigenvalues. The interpretation in terms
of homogeneous spaces of unitary groups is also discussed.
