Electronic level degeneracy in nonsymmorphic periodic or aperiodic crystals

The theory of space-group representations is extended to aperiodic crystals by reformulating it as the theory of symmetry-required degeneracies of electronic levels that emerges from the Fourier-space approach to crystal symmetry. As an illustration it is shown that the nonvanishing of a simple linear combination of phase functions belonging to commuting elements from the little group of q requires the degeneracy of all levels with generalized Bloch wave vector q. This condition is applied to all cubic and icosahedral centrosymmetric nonsymmorphic space groups, and to the two nonsymmorphic space groups of periodic crystals that have no systematic extinctions.