On the Symmetries of Spherical Harmonics

This paper extends the work described in a previous paper by one of the authors (Altmann 1957). The spherical harmonics that belong to the irreducible representations of the cubic groups are now given up to and including l = 12. Also, for all point groups the expansions in spherical harmonics that are given belong to the separate columns of the irreducible representations (whereas before they were linear combinations of such functions). Accordingly, full tables for the irreducible representations for all crystallographic point groups are required and are given in the paper. Finally, a technique is described, and used throughout in the tables, to orthogonalize several expansions that belong to the same column of the same irreducible representation. Therefore, the different expansions listed in the tables are always fully orthogonal.