  The crystal and magnetic structures of single-crystalline hexagonal LuFeO$_3$
films have been studied using x-ray, electron and neutron diffraction methods.
The polar structure of these films are found to persist up to 1050 K; and the
switchability of the polar behavior is observed at room temperature, indicating
ferroelectricity. An antiferromagnetic order was shown to occur below 440 K,
followed by a spin reorientation resulting in a weak ferromagnetic order below
130 K. This observation of coexisting multiple ferroic orders demonstrates that
hexagonal LuFeO$_3$ films are room-temperature multiferroics.
