Induced Representations of Space Groups in q-basis

Any subgroup rep can generate some induced rep of a group (see Sect. 3.2.1). In the particular case of a space group G the small irreps of the little group Gk C G induce its full irreps. 

In this and the next subsections we consider induced reps of space groups generated by the irreps (3 of the site-symmetry subgroup Gq of point q in the direct lattice space (/ 1 G). This type of induced rep of space groups was considered in [13]. In [39] a more general concept of band reps was introduced. All elements of the space group G can be written in the form VB+a , where Vr is a fractional lattice translation associated with the F point-group element R and a is a direct lattice translation vector. 

Let the local functions Wj f r) = W (r — q), (i = 1, 2. n ) be centered at point q of the direct lattice and span the space of the irrep / of the site-symmetry group Gq C G with matrices d gq) and characters X H9q) 9q Gq) The nature of these functions depends on the physical problem under consideration. In the electron-band theory of crystals lF) (r — qA) are atomic functions of atom A. In phonon spectroscopy applications — qA) mean the components of atomic displacements of 

The local functions associated with the smallest unit cell are obtained from local 

Functions in other unit cells are obtained through 

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