Tensor fields on crystals

A new method is presented to determine the irreducible representations of the space group of a crystal contained in the representation whose basis functions are the components of a tensor field defined on the atoms of a crystal. This reducible representation is the direct product of a tensor representation, dependent only on the tensor, and a permutation representation dependent only on how the atoms permute under elements of the space group. The permutation representation is first separately reduced prior to the reduction of the direct product. The permutation representation is shown to be an induced representation and its reduction is facilitated using the theory of induced representations. Examples and tables of results of applying this method are given in the case of a polar vector tensor field, applicable to lattice vibrational problems, and crystals, as the diamond structure, of space group symmetry O_h⁷.