International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.5, pp. 168176
Section 1.5.5. Examples and conclusions^{a}Faculty of Physics, University of Sofia, bulv. J. Boucher 5, 1164 Sofia, Bulgaria , and ^{b}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany 
In this section, four examples are considered in each of which the crystallographic classification scheme for the irreps is compared with the traditional one:^{5}

The asymmetric units of IT A are displayed in Figs. 1.5.5.1 to 1.5.5.4 by dashed lines. In Tables 1.5.5.1 to 1.5.5.4, the kvector types of CDML are compared with the Wintgen (Wyckoff) positions of IT A. The parameter ranges are chosen such that each star of k is represented exactly once. Sets of symmetry points, lines or planes of CDML which belong to the same Wintgen position are separated by horizontal lines in Tables 1.5.5.1 to 1.5.5.3. The uniarm description is listed in the last entry of each Wintgen position in Tables 1.5.5.1 and 1.5.5.2. In Table 1.5.5.4, so many kvector types of CDML belong to each Wintgen position that the latter are used as headings under which the CDML types are listed.

In addition, in the transition from a holosymmetric space group to a nonholosymmetric space group , the order of the little cogroup of a special k vector of may be reduced in . Such a k vector may then be incorporated into a more general Wintgen position of and described by an extension of the parameter range.
Example
Plane : In , see Fig. 1.5.5.1, all points and lines of the boundary of the asymmetric unit are special. In , see Fig. 1.5.5.2, the lines Δ and (∼ means equivalent) are special but Σ, G and belong to the plane . The free parameter range on the line is of the full parameter range of , see Section 1.5.5.3. Therefore, the parameter ranges of in x, y, 0 can be taken as: for and (for Σ) .
Is it easy to recognize those letters of CDML which belong to the same Wintgen position? In , the lines Λ and V (V exists for only) are parallel, as are Σ and F, but the lines Y and U are not (F and U exist for only). The planes and (D for only) are parallel but the planes and are not. Nevertheless, each of these pairs belongs to one Wintgen position, i.e. describes one type of k vector.
For the uniarm description of a Wintgen position it is easy to check whether the parameter ranges for the general or special constituents of the representation domain or asymmetric unit have been stated correctly. For this purpose one may define the field of k as the parameter space (point, line, plane or space) of a Wintgen position. For the check, one determines that part of the field of k which is inside the unit cell. The order of the little cogroup ( represents those operations which leave the field of k fixed pointwise) is divided by the order of the stabilizer [which is the set of all symmetry operations (modulo integer translations) that leave the field invariant as a whole]. The result gives the independent fraction of the abovedetermined volume of the unit cell or the area of the plane or length of the line.
If the description is not uniarm, the uniarm parameter range will be split into the parameter ranges of the different arms. The parameter ranges of the different arms are not necessarily equal; see the second of the following examples.
Examples

As has been shown, IT A can serve as a basis for the classification of irreps of space groups by using the concept of reciprocalspace groups:

In principle, both approaches are equivalent: the traditional one by Brillouin zone, basic domain and representation domain, and the crystallographic one by unit cell and asymmetric unit of IT A. Moreover, it is not difficult to relate one approach to the other, see the figures and Tables 1.5.5.1 to 1.5.5.4. The conclusions show that the crystallographic approach for the description of irreps of space groups has several advantages as compared to the traditional approach. Owing to these advantages, CDML have already accepted the crystallographic approach for triclinic and monoclinic space groups. However, the advantages are not restricted to such low symmetries. In particular, the simple boundary conditions and shapes of the asymmetric units result in simple equations for the boundaries and shapes of volume elements, and facilitate numerical calculations, integrations etc. If there are special reasons to prefer k vectors inside or on the boundary of the Brillouin zone to those outside, then the advantages and disadvantages of both approaches have to be compared again in order to find the optimal method for the solution of the problem.
The crystallographic approach may be realized in three different ways:

References
Davies, B. L. & Dirl, R. (1987). Various classification schemes for irreducible space group representations. In Proceedings of the 15th international colloquium on grouptheoretical methods in physics, edited by R. Gilmore, pp. 728–733. Singapore: World Scientific.Stokes, H. T., Hatch, D. M. & Nelson, H. M. (1993). Landau, Lifshitz, and weak Lifshitz conditions in the Landau theory of phase transitions in solids. Phys. Rev. B, 47, 9080–9083.