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. 162188
doi: 10.1107/97809553602060000553 Chapter 1.5. Crystallographic viewpoints in the classification of spacegroup representations^{a}Faculty of Physics, University of Sofia, bulv. J. Boucher 5, 1164 Sofia, Bulgaria , and ^{b}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany The k vectors are vectors in reciprocal space and play an important role in the description of spacegroup representations. Chapter 1.5 deals with the classification of these k vectors with special regard to crystallographic points of view. In 1941, Wintgen found that the k vectors of any space group can be classified in a natural way analogous to the classification of the Wyckoff positions of the symmorphic space groups. This is possible by introducing the socalled reciprocalspace group, which is isomorphic to a symmorphic space group. The symmetry types of k vectors correspond to the Wyckoff positions of this symmorphic space group and the tables of the Wyckoff positions in Volume A of International Tables for Crystallography present the classification directly. In this chapter, the basic concepts of representations of space groups are defined and the reciprocalspace group is introduced. The sometimes complicated Brillouin zone and its representation domain may be replaced by the often much simpler conventional unit cell of the reciprocal lattice and its asymmetric unit. The different k vectors of the same symmetry type are characterized by parameters which correspond to the coordinates of the representative points of the Wyckoff positions. The ranges of these parameters are chosen in such a way that each k vector is listed exactly once in the kvector table of the space group. The Wintgen classification is applied in several examples (space groups , , , , , , , , and ) and compared with the usual classification. 

This new chapter on representations widens the scope of the general topics of reciprocal space treated in this volume.
Spacegroup representations play a growing role in physical applications of crystal symmetry. They are treated in a number of papers and books but comparison of the terms and the listed data is difficult. The main reason for this is the lack of standards in the classification and nomenclature of representations. As a result, the reader is confronted with different numbers of types and barely comparable notations used by the different authors, see e.g. Stokes & Hatch (1988), Table 7.
The k vectors, which can be described as vectors in reciprocal space, play a decisive role in the description and classification of spacegroup representations. Their symmetry properties are determined by the socalled reciprocalspace group which is always isomorphic to a symmorphic space group . The different symmetry types of k vectors correspond to the different kinds of point orbits in the symmorphic space groups . The classification of point orbits into Wyckoff positions in International Tables for Crystallography Volume A (IT A) (2005) can be used directly to classify the irreducible representations of a space group, abbreviated irreps; the Wyckoff positions of the symmorphic space groups form a basis for a natural classification of the irreps. This was first discovered by Wintgen (1941). Similar results have been obtained independently by Raghavacharyulu (1961), who introduced the term reciprocalspace group. In this chapter a classification of irreps is provided which is based on Wintgen's idea.
Although this idea is now more than 50 years old, it has been utilized only rarely and has not yet found proper recognition in the literature and in the existing tables of spacegroup irreps. Slater (1962) described the correspondence between the special k vectors of the Brillouin zone and the Wyckoff positions of space group . Similarly, Jan (1972) compared Wyckoff positions with points of the Brillouin zone when describing the symmetry of the Fermi surface for the pyrite structure. However, the widespread tables of Miller & Love (1967), Zak et al. (1969), Bradley & Cracknell (1972) (abbreviated as BC), Cracknell et al. (1979) (abbreviated as CDML), and Kovalev (1986) have not made use of this kind of classification and its possibilities, and the existing tables are unnecessarily complicated, cf. Boyle (1986).
In addition, historical reasons have obscured the classification of irreps and impeded their application. The first considerations of irreps dealt only with space groups of translation lattices (Bouckaert et al., 1936). Later, other space groups were taken into consideration as well. Instead of treating these (lower) symmetries as such, their irreps were derived and classified by starting from the irreps of lattice space groups and proceeding to those of lower symmetry. This procedure has two consequences:
The use of the reciprocalspace group avoids both these detours.
In this chapter we consider in more detail the reciprocalspacegroup approach and show that widely used crystallographic conventions can be adopted for the classification of spacegroup representations. Some basic concepts are developed in Section 1.5.3. Possible conventions are discussed in Section 1.5.4. The consequences and advantages of this approach are demonstrated and discussed using examples in Section 1.5.5.
The aim of this section is to give a brief overview of some of the basic concepts related to groups and their representations. Its content should be of some help to readers who wish to refresh their knowledge of space groups and representations, and to familiarize themselves with the kind of description in this chapter. However, it can not serve as an introductory text for these subjects. The interested reader is referred to books dealing with spacegroup theory, representations of space groups and their applications in solidstate physics: see Bradley & Cracknell (1972) or Chapter 1.2 of IT D (Physical properties of crystals) by Janssen (2003).
Group theory is the proper tool for studying symmetry in science. The elements of the crystallographic groups are rigid motions (isometries) with regard to performing one after another. The set of all isometries that map an object onto itself always fulfils the group postulates and is called the symmetry or the symmetry group of that object; the isometry itself is called a symmetry operation. Symmetry groups of crystals are dealt with in this chapter. In addition, groups of matrices with regard to matrix multiplication (matrix groups) are considered frequently. Such groups will sometimes be called realizations or representations of abstract groups.
Many applications of group theory to physical problems are closely related to representation theory, cf. Rosen (1981) and references therein. In this section, matrix representations Γ of finite groups are considered. The concepts of homomorphism and matrix groups are of essential importance.
A group is a homomorphic image of a group if there exists a mapping of the elements of onto the elements of that preserves the multiplication relation (in general several elements of are mapped onto one element of ): if and , then holds for all elements of and (the image of the product is equal to the product of the images). In the special case of a onetoone mapping, the homomorphism is called an isomorphism.
A matrix group is a group whose elements are nonsingular square matrices. The law of combination is matrix multiplication and the group inverse is the inverse matrix. In the following we will be concerned with some basic properties of finite matrix groups relevant to representations.
Let and be two matrix groups whose matrices are of the same dimension. They are said to be equivalent if there exists a (nonsingular) matrix S such that holds. Equivalence implies isomorphism but the inverse is not true: two matrix groups may be isomorphic without being equivalent. According to the theorem of SchurAuerbach, every finite matrix group is equivalent to a unitary matrix group (by a unitary matrix group we understand a matrix group consisting entirely of unitary matrices).
A matrix group is reducible if it is equivalent to a matrix group in which every matrix M is of the form see e.g. Lomont (1959), p. 47. The group is completely reducible if it is equivalent to a matrix group in which for all matrices R the submatrices X are O matrices (consisting of zeros only). According to the theorem of Maschke, a finite matrix group is completely reducible if it is reducible. A matrix group is irreducible if it is not reducible.
A (matrix) representation of a group is a homomorphic mapping of onto a matrix group . In a representation Γ every element is associated with a matrix . The dimension of the matrices is called the dimension of the representation.
The abovementioned theorems on finite matrix groups can be applied directly to representations: we can restrict the considerations to unitary representations only. Further, since every finite matrix group is either completely reducible into irreducible constituents or irreducible, it follows that the infinite set of all matrix representations of a group is known in principle once the irreducible representations are known. Naturally, the question of how to construct all nonequivalent irreducible representations of a finite group and how to classify them arises.
Linear representations are especially important for applications. In this chapter only linear representations of space groups will be considered. Realizations and representations are homomorphic images of abstract groups, but not all of them are linear. In particular, the action of space groups on point space is a nonlinear realization of the abstract space groups because isometries and thus symmetry operations of space groups are nonlinear operations. The same holds for their description by matrixcolumn pairs (W, w),^{1} by the general position, or by augmented matrices, see IT A, Part 8 . Therefore, the isomorphic matrix representation of a space group, mostly used by crystallographers and listed in the spacegroup tables of IT A as the general position, is not linear.
In crystallography one deals with real crystals. In many cases the treatment of the crystal is much simpler, but nevertheless describes the crystal and its properties very well, if the real crystal is replaced by an `ideal crystal'. The real crystal is then considered to be a finite piece of an undisturbed, periodic, and thus infinitely extended arrangement of particles or their centres: ideal crystals are periodic objects in threedimensional point space , also called direct space. Periodicity means that there are translations among the symmetry operations of ideal crystals. The symmetry group of an ideal crystal is called its space group .
Space groups are of special interest for our problem because:
Therefore, space groups are introduced here in a slightly more detailed manner than the other concepts. In doing this we follow the definitions and symbolism of IT A, Part 8 .
To each space group belongs an infinite set of translations, the translation lattice of . The lattice forms an infinite Abelian invariant subgroup of . For each translation its translation vector is defined. The set of all translation vectors is called the vector lattice L of . Because of the finite size of the atoms constituting the real crystal, the lengths of the translation vectors of the ideal crystal cannot be arbitrarily small; rather there is a lower limit for their length in the range of a few Å.
When referred to a coordinate system , consisting of an origin O and a basis , the elements , i.e. the symmetry operations of the space group , are described by matrixcolumn pairs (W, w) with matrix part W and column part w. The translations of are represented by pairs , where I is the unit matrix and is the column of coefficients of the translation vector . The basis can always be chosen such that all columns and no other columns of translations consist of integers. Such a basis is called a primitive basis. For each vector lattice L there exists an infinite number of primitive bases.
The space group can be decomposed into left cosets relative to : The coset representatives form the finite set , with , where o is the column consisting of zeros only. The factor group is isomorphic to the point group of (called in books on representation theory) describing the symmetry of the external shape of the macroscopic crystal and being represented by the matrices . If V can be chosen such that all , then is called a symmorphic space group . A symmorphic space group can be recognized easily from its conventional Hermann–Mauguin symbol which does not contain any screw or glide component. In terms of group theory, a symmorphic space group is the semidirect product of and , cf. BC, p. 44. In symmorphic space groups (and in no others) there are sitesymmetry groups which are isomorphic to the point group of .
Space groups can be classified into 219 (affine) spacegroup types either by isomorphism or by affine equivalence; the 230 crystallographic spacegroup types are obtained by restricting the transformations available for affine equivalence to those with positive determinant, cf. IT A, Section 8.2.1. Many important properties of space groups are shared by all space groups of a type. In such a case one speaks of properties of the type. For example, if a space group is symmorphic, then all space groups of its type are symmorphic, so that one normally speaks of a symmorphic spacegroup type.
With the concept of symmorphic space groups one can also define the arithmetic crystal classes: Let be a symmorphic space group referred to a primitive basis and its set of coset representatives with for all columns. To all those space groups can be assigned for which a primitive basis can be found such that the matrix parts of their sets V are the same as those of , only the columns may differ. In this way, to a type of symmorphic space groups , other types of space groups are assigned, i.e. the spacegroup types are classified according to the symmorphic spacegroup types. These classes are called arithmetic crystal classes of space groups or of spacegroup types.
There are 73 arithmetic crystal classes corresponding to the 73 types of symmorphic space groups; between 1 and 16 spacegroup types belong to an arithmetic crystal class. A matrixalgebraic definition of arithmetic crystal classes and a proposal for their nomenclature can be found in IT A, Section 8.2.3 ; see also Section 8.3.4 and Table 8.3.4.1 .
For representation theory we follow the terminology of BC and CDML.
Let be referred to a primitive basis. For the following, the infinite set of translations, based on discrete cyclic groups of infinite order, will be replaced by a (very large) finite set in the usual way. One assumes the Born–von Karman boundary conditions to hold, where , (0, 1, 0) or (0, 0, 1) and is a large integer for i = 1, 2 or 3, respectively. Then for any lattice translation (I, t), holds, where Nt is the column . If the (infinitely many) translations mapped in this way onto (I, o) form a normal subgroup of , then the mapping described by (1.5.3.3) is a homomorphism. There exists a factor group of relative to with translation subgroup which is finite and is sometimes called the finite space group.
Only the irreducible representations (irreps) of these finite space groups will be considered. The definitions of spacegroup type, symmorphic space group etc. can be transferred to these groups. Because is Abelian, is also Abelian. Replacing the space group by means that the especially well developed theory of representations of finite groups can be applied, cf. Lomont (1959), Jansen & Boon (1967). For convenience, the prime ′ will be omitted and the symbol will be used instead of ; will be denoted by in the following.
Because (formerly ) is Abelian, its irreps are onedimensional and consist of (complex) roots of unity. Owing to equations (1.5.3.2) and (1.5.3.3), the irreps of have the form where t is the column , , and and are integers.
Given a primitive basis of L, mathematicians and crystallographers define the basis of the dual or reciprocal lattice by where is the scalar product between the vectors and is the unit matrix (see e.g. Chapter 1.1, Section 1.1.3 ). Texts on the physics of solids redefine the basis of the reciprocal lattice , lengthening each of the basis vectors by the factor . Therefore, in the physicist's convention the relation between the bases of direct and reciprocal lattice reads (cf. BC, p. 86): In the present chapter only the physicist's basis of the reciprocal lattice is employed, and hence the use of should not lead to misunderstandings. The set of all vectors K,^{2} integer, is called the lattice reciprocal to L or the reciprocal lattice .^{3}
If one adopts the notation of IT A, Part 5, the basis of direct space is denoted by a row , where means transposed. For reciprocal space, the basis is described by a column .
To each lattice generated from a basis a reciprocal lattice is generated from the basis . Both lattices, L and , can be compared most easily by referring the direct lattice L to its conventional basis as defined in Chapters 2.1 and 9.1 of IT A. In this case, the lattice L may be primitive or centred. If forms a primitive basis of L, i.e. if L is primitive, then the basis forms a primitive basis of . If L is centred, i.e. is not a primitive basis of L, then there exists a centring matrix P, , by which three linearly independent vectors of L with rational coefficients are generated from those with integer coefficients, cf. IT A, Table 5.1.3.1 .
Moreover, P can be chosen such that the set of vectors forms a primitive basis of L. Then the basis vectors of the lattice reciprocal to the lattice generated by are determined by and form a primitive basis of .
Because , not all vectors K of the form (1.5.3.7) belong to . If are the (integer) coefficients of these vectors K referred to and are the vectors of , then is a vector of if and only if the coefficients are integers. In other words, has to fulfil the equation
As is well known, the Bravais type of the reciprocal lattice is not necessarily the same as that of its direct lattice L. If W is the matrix of a (point) symmetry operation of the direct lattice, referred to its basis , then is the matrix of the same symmetry operation of the reciprocal lattice but referred to the dual basis . This does not affect the symmetry because in a (symmetry) group the inverse of each element in the group also belongs to the group. Therefore, the (point) symmetries of a lattice and its reciprocal lattice are always the same. However, there may be differences in the matrix descriptions due to the different orientations of L and relative to the symmetry elements of and due to the reference to the different bases and . For example, if L has the point symmetry (Hermann–Mauguin symbol) , then the symbol for the point symmetry of is and vice versa.
Let be a conventional basis of the lattice L of the space group . From (1.5.3.6), and , equation (1.5.3.4) can be written Equation (1.5.3.12) has the same form if a primitive basis of L has been chosen. In this case, the vector k is given by .
Let a primitive basis be chosen for the lattice L. The set of all vectors k (known as wavevectors) forms a discontinuous array. Consider two wavevectors k and , where K is a vector of the reciprocal lattice . Obviously, k and describe the same irrep of . Therefore, to determine all irreps of it is necessary to consider only the wavevectors of a small region of the reciprocal space, where the translation of this region by all vectors of fills the reciprocal space without gap or overlap. Such a region is called a fundamental region of . (The nomenclature in literature is not quite uniform. We follow here widely adopted definitions.)
The fundamental region of is not uniquely determined. Two types of fundamental regions are of interest in this chapter:
Let k be some vector according to (1.5.3.12) and W be the matrices of . The following definitions are useful:
Definition. The set of all matrices for which forms a group which is called the little cogroup of k. The vector k is called general if ; otherwise and k is called special.
The little cogroup is a subgroup of the point group . Consider the coset decomposition of relative to .
Definition. If is a set of coset representatives of relative to , then the set is called the star of k and the vectors are called the arms of the star.
The number of arms of the star of k is equal to the order of the point group divided by the order of the symmetry group of k. If k is general, then there are vectors from the orbit of k in each fundamental region and arms of the star. If k is special with little cogroup , then the number of arms of the star of k and the number of k vectors in the fundamental region from the orbit of k is .
Equation (1.5.3.14) for k resembles the equation by which the fixed points of the symmetry operation of a symmorphic space group are determined. Indeed, the orbits of k defined by (1.5.3.13) correspond to the point orbits of , the little cogroup of k corresponds to the sitesymmetry group of that point X whose coordinates have the same values as the vector coefficients of k, and the star of k corresponds to a set of representatives of X in . (The analogue of the little group is rarely considered in crystallography.)
All symmetry operations of may be obtained as combinations of an operation that leaves the origin fixed with a translation of L, i.e. are of the kind . We now define the analogous group for the k vectors. Whereas is a realization of the corresponding abstract group in direct (point) space, the group to be defined will be a realization of it in reciprocal (vector) space.
Definition. The group which is the semidirect product of the point group and the translation group of the reciprocal lattice of is called the reciprocalspace group of .
The elements of are the operations with and . In order to emphasize that is a group acting on reciprocal space and not the inverse of a space group (whatever that may mean) we insert a hyphen `' between `reciprocal' and `space'.
From the definition of it follows that space groups of the same type define the same type of reciprocalspace group . Moreover, as does not depend on the column parts of the spacegroup operations, all space groups of the same arithmetic crystal class determine the same type of ; for arithmetic crystal class see Section 1.5.3.2. Following Wintgen (1941), the types of reciprocalspace groups are listed for the arithmetic crystal classes of space groups, i.e. for all space groups , in Appendix 1.5.1.
Because of the isomorphism between the reciprocalspace groups and the symmorphic space groups one can introduce crystallographic conventions in the classification of spacegroup irreps. These conventions will be compared with those which have mainly been used up to now. Illustrative examples to the following more theoretical considerations are discussed in Section 1.5.5.1.
Different types of regions of reciprocal space may be chosen as fundamental regions, see Section 1.5.3.4. The most frequently used type is the first Brillouin zone, which is the Wigner–Seitz cell (or Voronoi region, Dirichlet domain, domain of influence; cf. IT A, Chapter 9.1 ) of the reciprocal lattice. It has the property that with each k vector also its star belongs to the Brillouin zone. Such a choice has three advantages:
Of these advantages only the third may be essential. For the classification of irreps the minimal domains, see Section 1.5.4.2, are much more important than the fundamental regions. The minimal domain does not display the pointgroup symmetry anyway and the distinguished k vectors always belong to its boundary however the minimal domain may be chosen.
The serious disadvantage of the Brillouin zone is its often complicated shape which, moreover, depends on the lattice parameters of . The body that represents the Brillouin zone belongs to one of the five Fedorov polyhedra (more or less distorted versions of the cubic forms cube, rhombdodecahedron or cuboctahedron, of the hexagonal prism, or of the tetragonal elongated rhombdodecahedron). A more detailed description is that by the 24 symmetrische Sorten (Delaunay sorts) of Delaunay (1933), Figs. 11 and 12. According to this classification, the Brillouin zone may display three types of polyhedra of cubic, one type of hexagonal, two of rhombohedral, three of tetragonal, six of orthorhombic, six of monoclinic, and three types of triclinic symmetry.
For low symmetries the shape of the Brillouin zone is so variable that BC, p. 90 ff. chose a primitive unit cell of for the fundamental regions of triclinic and monoclinic crystals. This cell also reflects the point symmetry of , it has six faces only, and although its shape varies with the lattice constants all cells are affinely equivalent. For space groups of higher symmetry, BC and most other authors prefer the Brillouin zone.
Considering as a lattice, one can refer it to its conventional crystallographic lattice basis. Referred to this basis, the unit cell of is always an alternative to the Brillouin zone. With the exception of the hexagonal lattice, the unit cell of reflects the point symmetry, it has only six faces and its shape is always affinely equivalent for varying lattice constants. For a space group with a primitive lattice, the abovedefined conventional unit cell of is also primitive. If has a centred lattice, then also belongs to a type of centred lattice and the conventional cell of [not to be confused with the cell spanned by the basis dual to the basis ] is larger than necessary. However, this is not disturbing because in this context the fundamental region is an auxiliary construction only for the definition of the minimal domain; see Section 1.5.4.2.
One can show that all irreps of can be built up from the irreps of . Moreover, to find all irreps of it is only necessary to consider one k vector from each orbit of k, cf. CDML, p. 31.
Definition. A simply connected part of the fundamental region which contains exactly one k vector of each orbit of k is called a minimal domain Φ.
The choice of the minimal domain is by no means unique. One of the difficulties in comparing the published data on irreps of space groups is due to the different representation domains found in the literature.
The number of k vectors of each general k orbit in a fundamental region is always equal to the order of the point group of ; see Section 1.5.3.4. Therefore, the volume of the minimal domain Φ in reciprocal space is of the volume of the fundamental region. Now we can restrict the search for all irreps of to the k vectors within a minimal domain Φ.
In general, in representation theory of space groups the Brillouin zone is taken as the fundamental region and Φ is called a representation domain.^{4} Again, the volume of a representation domain in reciprocal space is of the volume of the Brillouin zone. In addition, as the Brillouin zone contains for each k vector all k vectors of the star of k, by application of all symmetry operations to Φ one obtains the Brillouin zone; cf. BC, p. 147. As the Brillouin zone may change its geometrical type depending on the lattice constants, the type of the representation domain may also vary with varying lattice constants; see examples (3) and (4) in Section 1.5.5.1.
The simplest crystal structures are the latticelike structures that are built up of translationally equivalent points (centres of particles) only. For such a structure the point group of the space group is equal to the point group of its lattice L. Such point groups are called holohedral, the space group is called holosymmetric. There are seven holohedral point groups of three dimensions: and . For the nonholosymmetric space groups holds.
In books on representation theory of space groups, holosymmetric space groups play a distinguished role. Their representation domains are called basic domains Ω. For holosymmetric space groups holds. If is nonholosymmetric, i.e. holds, Ω is defined by and is smaller than the representation domain Φ by a factor which is equal to the index of in . In the literature these basic domains are considered to be of primary importance. In Miller & Love (1967) only the irreps for the k vectors of the basic domains Ω are listed. Section 5.5 of BC and Davies & Cracknell (1976) state that such a listing is not sufficient for the nonholosymmetric space groups because . Section 5.5 of BC shows how to overcome this deficiency; Chapter 4 of CDML introduces new types of k vectors for the parts of Φ not belonging to Ω.
The crystallographic analogue of the representation domain in direct space is the asymmetric unit, cf. IT A. According to its definition it is a simply connected smallest part of space from which by application of all symmetry operations of the space group the whole space is exactly filled. For each spacegroup type the asymmetric units of IT A belong to the same topological type independent of the lattice constants. They are chosen as `simple' bodies by inspection rather than by applying clearly stated rules. Among the asymmetric units of the 73 symmorphic spacegroup types there are 31 parallelepipeds, 27 prisms (13 trigonal, 6 tetragonal and 8 pentagonal) for the noncubic, and 15 pyramids (11 trigonal and 4 tetragonal) for the cubic .
The asymmetric units of IT A – transferred to the groups of reciprocal space – are alternatives for the representation domains of the literature. They are formulated as closed bodies. Therefore, for inner points k, the asymmetric units of IT A fulfil the condition that each star of k is represented exactly once. For the surface, however, these conditions either have to be worked out or one gives up the condition of uniqueness and replaces exactly by at least in the definition of the minimal domain (see preceding footnote^{4}). The examples of Section 1.5.5.1 show that the conditions for the boundary of the asymmetric unit and its special points, lines and planes are in many cases much easier to formulate than those for the representation domain.
The kvector coefficients. For each k vector one can derive a set of irreps of the space group . Different k vectors of a k orbit give rise to equivalent irreps. Thus, for the calculation of the irreps of the space groups it is essential to identify the orbits of k vectors in reciprocal space. This means finding the sets of all k vectors that are related by the operations of the reciprocalspace group according to equation (1.5.3.13). The classification of these k orbits can be done in analogy to that of the point orbits of the symmorphic space groups, as is apparent from the comparison of equations (1.5.3.14) and (1.5.3.15).
The classes of point orbits in direct space under a space group are well known and are listed in the spacegroup tables of IT A. They are labelled by Wyckoff letters. The stabilizer of a point X is called the sitesymmetry group of X, and a Wyckoff position consists of all orbits for which the sitesymmetry groups are conjugate subgroups of . Let be a symmorphic space group . Owing to the isomorphism between the reciprocalspace groups and the symmorphic space groups , the complete list of the types of special k vectors of is provided by the Wyckoff positions of . The groups and correspond to each other and the multiplicity of the Wyckoff position (divided by the number of centring vectors per unit cell for centred lattices) equals the number of arms of the star of k. Let the vectors t of L be referred to the conventional basis of the spacegroup tables of IT A, as defined in Chapters 2.1 and 9.1 of IT A. Then, for the construction of the irreducible representations of the coefficients of the k vectors must be referred to the basis of reciprocal space dual to in direct space. These kvector coefficients may be different from the conventional coordinates of listed in the Wyckoff positions of IT A.
Example
Let be a space group with an Icentred cubic lattice L, conventional basis . Then is an Fcentred lattice. If referred to the conventional basis with , the k vectors with coefficients 1 0 0, 0 1 0 and 0 0 1 do not belong to due to the `extinction laws' well known in Xray crystallography. However, in the standard basis of , isomorphic to , the vectors 1 0 0, 0 1 0 and 0 0 1 point to the vertices of the facecentred cube and thus correspond to 2 0 0, 0 2 0 and 0 0 2 referred to the conventional basis .
In the following, three bases and, therefore, three kinds of coefficients of k will be distinguished:
The relations between conventional and adjusted coefficients are listed for the different Bravais types of reciprocal lattices in Table 1.5.4.1, and those between adjusted and primitive coordinates in Table 1.5.4.2. If adjusted coefficients are used, then IT A is as suitable for dealing with irreps as it is for handling spacegroup symmetry.


In order to avoid confusion, in the following the analogues to the Wyckoff positions of will be called Wintgen positions of ; the coordinates of the Wyckoff position are replaced by the kvector coefficients of the Wintgen position, the Wyckoff letter will be called the Wintgen letter, and the symbols for the site symmetries of are to be read as the symbols for the little cogroups of the k vectors in . The multiplicity of a Wyckoff position is retained in the Wintgen symbol in order to facilitate the use of IT A for the description of symmetry in k space. However, it is equal to the multiplicity of the star of k only in the case of primitive lattices .
In analogy to a Wyckoff position, a Wintgen position is a set of orbits of k vectors. Each orbit as well as each star of k can be represented by any one of its k vectors. The zero, one, two or three parameters in the kvector coefficients define points, lines, planes or the full parameter space. The different stars of a Wintgen position are obtained by changing the parameters.
Remark. Because reciprocal space is a vector space, there is no origin choice and the Wintgen letters are unique (in contrast to the Wyckoff letters, which may depend on the origin choice). Therefore, the introduction of Wintgen sets in analogy to the Wyckoff sets of IT A, Section 8.3.2 is not necessary.
It may be advantageous to describe the different stars belonging to a Wintgen position in a uniform way. For this purpose one can define:
Definition. Two k vectors of a Wintgen position are uniarm if one can be obtained from the other by parameter variation. The description of the stars of a Wintgen position is uniarm if the k vectors representing these stars are uniarm.
For nonholosymmetric space groups the representation domain Φ is a multiple of the basic domain Ω. CDML introduced new letters for stars of k vectors in those parts of Φ which do not belong to Ω. If one can make a new k vector uniarm to some k vector of the basic domain Ω by an appropriate choice of Φ and Ω, one can extend the parameter range of this k vector of Ω to Φ instead of introducing new letters. It turns out that indeed most of these new letters are unnecessary. This restricts the introduction of new types of k vectors to the few cases where it is indispensible. Extension of the parameter range for k means that the corresponding representations can also be obtained by parameter variation. Such representations can be considered to belong to the same type. In this way a large number of superfluous kvector names, which pretend a greater variety of types of irreps than really exists, can be avoided (Boyle, 1986). For examples see Section 1.5.5.1.
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:

Acknowledgements
The authors wish to thank the editor of this volume, Uri Shmueli, for his patient support, for his encouragement and for his valuable help. They are grateful to the Chairman of the Commission on International Tables, Theo Hahn, for his interest and advice. The material in this chapter was first published as an article of the same title in Z. Kristallogr. (1995), 210, 243–254. We are indebted to R. Oldenbourg Verlag, Munich, Germany, for allowing us to reprint parts of this article.
Appendix A1.5.1
This table is based on Table 1 of Wintgen (1941).
In order to obtain the Hermann–Mauguin symbol of from that of , one replaces any screw rotations by rotations and any glide reflections by reflections. The result is the symmorphic space group assigned to . For most space groups , the reciprocalspace group is isomorphic to , i.e. and belong to the same arithmetic crystal class. In the following cases the arithmetic crystal classes of and are different, i.e. can not be obtained in this simple way:

References
Altmann, S. L. (1977). Induced representations in crystals and molecules. London: Academic Press.Bouckaert, L. P., Smoluchowski, R. & Wigner, E. P. (1936). Theory of Brillouin zones and symmetry properties of wave functions in crystals. Phys. Rev. 50, 58–67.
Boyle, L. L. (1986). The classification of space group representations. In Proceedings of the 14th international colloquium on grouptheoretical methods in physics, pp. 405–408. Singapore: World Scientific.
Boyle, L. L. & Kennedy, J. M. (1988). Raumgruppen Charakterentafeln. Z. Kristallogr. 182, 39–42.
Bradley, C. J. & Cracknell, A. P. (1972). The mathematical theory of symmetry in solids: representation theory for point groups and space groups. Oxford: Clarendon Press.
Cracknell, A. P., Davies, B. L., Miller, S. C. & Love, W. F. (1979). Kronecker product tables, Vol. 1, General introduction and tables of irreducible representations of space groups. New York: IFI/Plenum.
Davies, B. L. & Cracknell, A. P. (1976). Some comments on and addenda to the tables of irreducible representations of the classical space groups published by S. C. Miller and W. F. Love. Acta Cryst. A32, 901–903.
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.
Delaunay, B. (1933). Neue Darstellung der geometrischen Kristallographie. Z. Kristallogr. 84, 109–149; 85, 332. (In German.)
International Tables for Crystallography (2005). Vol. A. Spacegroup symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.
Jan, J.P. (1972). Space groups for Fermi surfaces. Can. J. Phys. 50, 925–927.
Jansen, L. & Boon, M. (1967). Theory of finite groups. Applications in physics: symmetry groups of quantum mechanical systems. Amsterdam: NorthHolland.
Janssen, T. (2003). International tables for crystallography, Vol. D, Physical properties of crystals, edited by A. Authier, ch. 1.2, Representations of crystallographic groups. Dordrecht: Kluwer Academic Publishers.
Kovalev, O. V. (1986). Irreducible and induced representations and corepresentations of Fedorov groups. Moscow: Nauka. (In Russian.)
Lomont, J. S. (1959). Applications of finite groups. New York: Academic Press.
Miller, S. C. & Love, W. F. (1967). Tables of irreducible representations of space groups and corepresentations of magnetic space groups. Boulder: Pruett Press.
Raghavacharyulu, I. V. V. (1961). Representations of space groups. Can. J. Phys. 39, 830–840.
Rosen, J. (1981). Resource letter SP2: Symmetry and group theory in physics. Am. J. Phys. 49, 304–319.
Slater, L. S. (1962). Quantum theory of molecules and solids, Vol. 2. Amsterdam: McGrawHill.
Stokes, H. T. & Hatch, D. M. (1988). Isotropy subgroups of the 230 crystallographic space groups. 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.
Wintgen, G. (1941). Zur Darstellungstheorie der Raumgruppen. Math. Ann. 118, 195–215. (In German.)
Zak, J., Casher, A., Glück, M. & Gur, Y. (1969). The irreducible representations of space groups. New York: Benjamin.