Band representations and symmetry types of bands in solids

Symmetry types of bands in solids are specified by means of band representations of space groups. This is a new kind of representation that corresponds to bands of energies rather than to single energies as in the case of usual representations. It is shown that each band representation defines a symmetry type of a band by specifying the symmetry of localized orbitals with respect to a whole lattice of point group centers. In this symmetry specification the quasicoordinate $\overrightarrow{\mathrm{q}}$ in the Wigner-Seitz cell plays a similar role to what is played by the quasimomentum $\overrightarrow{\mathrm{k}}$ in the symmetry specification of Bloch states in the Brillouin zone.