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angle angular momentum antisymmetric apply arbitrary atom axis basis vectors belonging Brillouin zone Chapter characters cible representations complex conjugate components configuration consequently consider contains corre corresponding cosets crystal decomposed decomposition defined denote determine diagonal direct product displacements eigenfunctions eigenvalue eigenvectors electron energy levels equal equivalent fact finite group follows given group H Hamiltonian hence identity representation infinitesimal matrices infinitesimal operators invariant sub-group inversion irreducible repre irreducible representations lattice vector linear combinations Lorentz group matrix elements mirror rotation molecule multiplication non-zero normal coordinates nuclei obtain orthogonal oscillator parameters particles permutation group perturbation point group problem quantum mechanics quantum number readily verified reducible representation matrices result rotation group Schroedinger equation selection rules sentations single-electron space group spin function spinor sponding suppose symmetry group tation tensor theorem tion transform in accordance unit element unit matrix unit vectors unitary wave function written Young's tableau zero