  Magnetic order of the manganese and rare-earth lattices according to
different symmetry representations is observed in multiferroic hexagonal (h-)
DyMnO$_3$ by optical second harmonic generation and neutron diffraction. The
incompatibility reveals that the 3d-4f coupling in the h-$R$MnO$_3$ system ($R$
= Sc, Y, In, Dy - Lu) is substantially less developed than commonly expected.
As a consequence, magnetoelectric coupling effects in this type of split-order
parameter multiferroic that were previously assigned to a pronounced 3d-4f
coupling have now to be scrutinized with respect to their origin.
