Symmetry and Analyticity of Energy Bands in Solids

We show that pairs of simple energy bands exist that have identical symmetries for their Bloch functions, but different symmetries for the corresponding Wannier functions. For finite crystals such pairs of bands belong to equivalent band representations. In the case of infinite crystals, we use the analyticity of simple bands to prove the inequivalence of their band representations. This means that the energies of such pairs of bands cannot coincide despite the identical symmetries of their Bloch functions.