Site Symmetry and Induced Representations of Space Groups

2 Site Symmetry and Induced Representations of Space Groups 

As an example we consider the correlation table for the point group (Table 

The subduction of the subgroup irreps by the group irrep can be considered as the reciprocal operation to the induction of the group reps by the irreps of the subgroup 

Let group G be decomposed into left cosets with respect to H  

To express the invariance of L under D(h)(h H) independently of a particular choice of basis we write 

---

Site Symmetry and Induced Representations of Space Groups Table 3.6. Induced irreps of the point group Td... 

As follows from the theory of BR, the LOs obtained without any preliminary symmetry analysis, are in fact to be bases of some representations (in the general case reducible) of site-symmetry groups of the Wyckoff positions of atoms and bonds, the preference has to be given also to BRs induced from site group of bonds. In this case, steps 2 and 3 have to be replaced by (2a) Generation maximally LOs in the space of the canonical Bloch valence states by any known procedure. (3a) A proper symmetrization of generated LOs according to the results of the symmetry analysis of step 1. 

---