Simple and Composite Induced Representations

The use of the q basis of induced reps allows one to introduce the concept of simple induced reps, that facilitates the analysis of all possible types of induced reps for a given space group. An induced rep is called simple if it is impossible to spht up the space of this rep into subspaces that are invariant under operators p (p e G) and are 

By definition, a composite induced rep is a direct sum of the simple ones. As a group rep, a simple induced rep is reducible, so we prefer to avoid the expression irreducible induced rep used in [39]. The term introduced in [40], elementary induced rep , is equivalent to the term simple induced rep used in this book. 

All simple induced reps may be generated by induction from the irreps of site-symmetry groups Ggi, of a relatively small number of q points forming the set Q in the Wigner-Seitz unit cell of the direct lattice. The set Q consists of 

1) all the inequivalent symmetry points of the Wigner-Seitz unit cell  

2) one representative point from all the inequivalent symmetry lines and symmetry planes that do not contain the symmetry points. 

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