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The left‐ (right‐) regular projective representation of a finite group G and the corresponding ’’projective’’ representation of the left‐group algebra are defined for a given standard factor system, and special features of these constructions are discussed. Starting from a given projective unitary irreducible representation of a normal (but not necessarily Abelian) subgroup N of G, we obtain by induction the matrix elements of the projective unitary irreducible representations of G, where the corresponding group algebra is used as aid. These considerations are of interest for the construction of projective unitary irreducible representations of little cogroups of nonsymmorphic space groups. For the present method allows us to construct, for a q lying on the ’’surface’’ of the Brillouin zone, these projective representations out from unitary irreducible representations belonging to q’s of ’’lower’’ symmetry. This method is used to determine for all little cogroups of the nonsymmorphic space group Pn3n complete sets of projective unitary irreducible representations.

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... However, the value of any physical observable, which can be expressed in terms of the CG coefficients, is invariant under a change of basis (Birman et al 1976). In the past few years much attention has been devoted to the theoretical problem of calculating CG coefficients (Litvin and Zak 1968, Card 1973c, Birman 1974a, Sakata 1974, Berenson and Birman 1975, van den Broek and Cornwell 1978, Dirl 1979a, c, 1981, 1982, Chen et a1 1983 and some hand calculations have been done on a few space groups (Berenson et a1 1975, Suffczynski and Kunert 1978, Dirl 1979d, Kunert and Suffczynski 1980, Kunert 1983. Dirl (1979aDirl ( , c, 1981Dirl ( , 1982 exploits the fact that the columns of the CG matrix may be seen as symmetry adapted vectors which may then be constructed by projection operator techniques. ...

... However, the value of any physical observable, which can be expressed in terms of the CG coefficients, is invariant under a change of basis (Birman et al 1976). In the past few years much attention has been devoted to the theoretical problem of calculating CG coefficients (Litvin and Zak 1968, Card 1973c, Birman 1974a, Sakata 1974, Berenson and Birman 1975, van den Broek and Cornwell 1978, Dirl 1979a, c, 1981, 1982, Chen et a1 1983 and some hand calculations have been done on a few space groups (Berenson et a1 1975, Suffczynski and Kunert 1978, Dirl 1979d, Kunert and Suffczynski 1980, Kunert 1983. Dirl (1979aDirl ( , c, 1981Dirl ( , 1982 exploits the fact that the columns of the CG matrix may be seen as symmetry adapted vectors which may then be constructed by projection operator techniques. ...

... In the past few years much attention has been devoted to the theoretical problem of calculating CG coefficients (Litvin and Zak 1968, Card 1973c, Birman 1974a, Sakata 1974, Berenson and Birman 1975, van den Broek and Cornwell 1978, Dirl 1979a, c, 1981, 1982, Chen et a1 1983 and some hand calculations have been done on a few space groups (Berenson et a1 1975, Suffczynski and Kunert 1978, Dirl 1979d, Kunert and Suffczynski 1980, Kunert 1983. Dirl (1979aDirl ( , c, 1981Dirl ( , 1982 exploits the fact that the columns of the CG matrix may be seen as symmetry adapted vectors which may then be constructed by projection operator techniques. This elegant method correctly deals with the 'multiplicity problem' which arises from the fact that space groups are not simply reducible. ...

The wavevector selection rules (WVSR) occurring in the reduction of Kronecker products of space group unirreps are classified, for convenience, into three types. For WVSR of type I, Dirl (1979) has shown that special solutions of the multiplicity problem always exist. For WVSR of type II, Dirl has given a simple criterion for the existence for special solutions of the multiplicity problem and the authors show that, for all 230 (single and double) space groups, the Miller and Love matrix unirreps satisfy this criterion. WVSR of type III will be considered in a subsequent paper.

The goal of this paper is to study the regular projective repre-sentations of a discrete group G on a Hilbert space H ([9]). We describe the commuting algebras of the right, respectively left regular projective representations and present the existence theorem for infinite tensor prod-ucts of projective representations proved by E. Bédos and R. Conti in [2].

Equivalence of induced representations for finite groups is considered in order to determine those equivalence classes of space group representations which are linked by complex conjugation.

A general method is given for obtaining Clebsch–Gordan coefficients for finite groups, by considering the columns of the Clebsch–Gordan matrices as G‐adapted vectors and by identifying the multiplicity index as special column indices of the Kronecker product. The matrix representations are assumed to be projective ones, however not necessarily belonging to equivalent factor systems.

A general method for calculating Clebsch-Gordan coefficients is applied to determine such coefficients for the nonsymmorphic space group Pn3n.

Clebsch-Gordan coefficients for the type II Shubnikov space group Pn3'n are calculated in terms of such coefficients for the unitary subgroup Pn3n.

Useful relations are derived which allow us to determine for every unirrep of nonsymmorphic space groups, which contain the inversion as a point group operation, the equivalent complex conjugate representation.

Irreducible representations of compact groups can be partitioned into three classes (character test +,0,−). This classification is the same for real, complex, and quaternionic representations and in all three cases a peculiar, type-adapted form of the representation matrices may be chosen (t reps). In this paper it is shown how to construct t reps of semidirect products G s g starting with t reps of G and t reps of some covering groups of subgroups of g. The advantage of using t reps shows up in that the factor system of the little cogroups is real in two of three cases and that real, complex, and quaternionic representations are obtained simultaneously. The method is specialized to direct products and generalized to induction from normal subgroups.

Coupling coefficients for projective representations of finite groups are determined quite generally either by means of a general projection procedure, or as a linear combination of their corresponding Clebsch-Gordan coefficients. There we demonstrate that coupling- and Clebsch-Gordan coefficients are uniquely connected by special Clebsch-Gordan coefficients up to well-determined numerical factors.

The multiplicity formula for nonsymmorphic space group representations is reinvestigated by using explicitly projective representations for the little cogroups Pq&drarr;~=Gq&drarr;/T. Thereby useful identities and relations concerning the wave vector selection rules are derived for various cases which may occur for the elements of the Brillouin zone. These relations allow for nearly all cases a closed expression for the multiplicity without reference to a special space group.

A so-called representation (rep) group G is introduced which is formed by all the |G| distinct operators (or matrices) of an abstract group G in a rep space L and which is an m-fold covering group of another abstract group g. G forms a rep of G. The rep group differs from an abstract group in that its elements are not linearly independent and thus the number n of its linearly independent class operators is less than its class number N. A systematic theory is established for the rep group based on Dirac's CSCO (complete set of commuting operators) approach in quantum mechanics. This theory also comprises the rep theory for abstract groups as a special case of m=1. Three kinds of CSCO, the CSCO-I, -II, and -III, are defined which are the analogies of J2, (J2,Jz), and (J2,Jz,J¯z), respectively, for the rotation group SO3, where J¯z is the component of angular momentum in the intrinsic frame. The primitive characters, the irreducible basis and Clebsch-Gordan coefficients, and the irreducible matrices of the rep group G in any subgroup symmetry adaptation can be found by solving the eigenequations of the CSCO-I, -II, and -III of G, respectively, in appropriate vector spaces. It is shown that the rep group G has only n instead of N inequivalent irreducible representations (irreps), which are just the allowable irreps of the abstract group G in the space L. Therefore, the construction of the irreps of G in L can be replaced by that of G. The labor involved in the construction of the irreps of the rep group G with order |G| is no more than that for the group g with order |g|=|G|m, and thus tremendous labor can be saved by working with the rep group G instead of the abstract group G. Based on the rep-group theory, a new approach to the space-group rep theory is proposed, which is distinguished by its simplicity and applicability. Corresponding to each little group G(k), there is a rep group G'k. The n inequivalent irreps of G'k are essentially just the acceptable irreps of the little group G(k). Consequently the construction of the irreps of G(k) is almost as easy as that of the little co-group G0(k). An easily programmable algorithm is established for computing the Clebsch-Gordan series and Clebsch-Gordan coefficients of a space group simultaneously.

A general method for calculating Clebsch-Gordan coefficients is applied in order to compute for typical examples of Pn3n those unitary matrices which connect coupling-with their corresponding Clebsch-Gordan coefficients.

Specific subgroup chain adaptations of representations and corresponding CG-matrices are discussed. Simple applicable analytic formulas are derived which allow one to express supergroup CG-matrices in terms of subgroup CG-matrices. The results are applied to space groups.

It is shown that to find Clebsch‐Gordan coefficients of space groups (both single and double), the representations of the groups of k alone are required. This is another example demonstrating the well‐accepted fact that in applications of space groups it is sufficient to know the representations of the groups of k. Final formulas are derived that enable the calculation of the Clebsch‐Gordan coefficients from the representations of the groups of k. As an example the spin‐orbit coupling in solids is considered.

The construction of the irreducible representations of single and double nonsymmorphic space groups is discussed. The proof is given that for any symmetry element where the nonsymmorphism plays a role there is a finite group of lowest order such that its irreducible representations engender all the allowable representations of the little group. For most high symmetry elements the order of this optimal factor group is only twice the order of the corresponding point group of the wave vector. The computational advantages of using this group instead of other known factor groups are stressed.

Zak [J. Math. Phys. 1, 165 (1960)] has given a method for constructing the irreducible representations of space groups which involves inducing the representations of the full group from those of an invariant subgroup. When a representation of the subgroup is self‐conjugate, Zak's prescription for the induction is subject to a restriction which makes it inapplicable in some practical applications of the method. This paper presents a general prescription for carrying out the induction from self‐conjugate representations.

A practical method for calculating Clebsch−Gordan coefficients for crystal space groups is presented. It is based on properties of the space group irreducible representations as induced from ray representations of subgroups. Using this method, we obtain all Clebsch−Gordan coefficients for a family of representations in a single calculation: For space groups, for a given triangle of stars *k, *k′, *k″, where *k ⊗ *k′ - *k″, the coefficients for all allowable little group representations l, l′, l″ are obtained. In the following paper this is applied to rocksalt O5h−Fm3m and diamond O7h−Fd3m space groups.

For compact topological groups (discrete or continuous) a basis of the group algebra is defined which consists of irreducible tensors only. This tensor basis is generally discussed and compared with similar constructions for finite groups and SU (2).

It is well known that if the interaction between electrons in a metal is neglected, the energy spectrum has a zonal structure. The problem of these "Brillouin zones" is treated here from the point of view of group theory. In this theory, a representation of the symmetry group of the underlying problem is associated with every energy value. The symmetry, in the present case, is the space group, and the main difference as compared with ordinary problems is that while in the latter the representations form a discrete manifold and can be characterized by integers (as e.g., the azimuthal quantum number), the representations of a space group form a continuous manifold, and must be characterized by continuously varying parameters. It can be shown that in the neighborhood of an energy value with a certain representation, there will be energy values with all the representations the parameters of which are close to the parameters of the original representation. This leads to the well-known result that the energy is a continuous function of the reduced wave vector (the components of which are parameters of the above-mentioned kind), but allows in addition to this a systematic treatment of the "sticking" together of Brillouin zones. The treatment is carried out for the simple cubic and the body-centered and face-centered cubic lattices, showing the different possible types of zones.

By using the method described in the previous paper, based on properties of space group irreducible representations as induced from ray representations of subgroups, Clebsch−Gordan coefficients are calculated for *X ⊗ *X in diamond O7h−Fd3m and rocksalt O5h−Fm3m structures. Tables of coefficients for these stars are presented. An example of explicit calculation of the coefficients is given for these symmorphic and nonsymmorphic groups with multiplicity included in the former.

The procedure for setting up projection operators to derive symmetrized wave functions in crystals is discussed. This includes a description of the projection operator method, discussion of irreducible representations, basis functions, and the multiplication table for space groups, and the application of these methods to setting up symmetrized plane waves, and symmetrized linear combinations of atomic orbitals, for use with the OPW and APW methods of approximating to electronic wave functions in crystals.

A method is developed to obtain the character tables of nonsymmorphic space groups. The method is based on the possibility of obtaining all the irreducible representations of a group, if one knows all the irreducible representations of its invariant subgroup of index 2 or 3. It turns out that all the space groups have an invariant subgroup of index 2 or 3.

A general discussion of the treatment of the asymmorphic space groups is
given, with special reference to the hexagonal close-packed lattice
(P63mmc), for which the irreducible representations are given
in full and the lattice harmonics listed for all l. The expansions
possess the same properties as those given for the cubic groups in the
preceding paper.

our papers [11] and [12] we shall abbreviate the term "continuous unitary representation of (~" to "representation of (~". If X is a proper closed subgroup of (~ whose representations are in a suitable sense "all known" one may pose the following two questions. (a) Which representations of 3( are the restrictions to it of irreducible representations of ~? (b) Given such a representation of 3( how can one construct all irreducible representations of (~ of which it is the restriction? When 3( is the identity subgroup question (a) has a trivial answer (apart from questions of dimension) and question (b) is essentially the same as that of determining all irreducible representations of (~? However, for other choices of 3(, questions (a) and (b) can furnish a useful breakdown of the problem of determining all irreducible representations of (~ into two more accessible components. It is the primary purpose of this paper to discuss questions (a) and (b) and their application to the determination of the representations of (~ in the special case in which 3( is normal. Actually we shall find it more convenient to deal with the slight variation in which we identify representations of 3( which are quasi equivalent in the sense defined on page 195 of [12]. For the special case in which 3( is not only normal but commutative and in which ~ is a semi direct product of 3( and (~/3( this program has been carried out in outline in our paper

Lecture Noles in Physics R.Dirl 2080 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at

- a P Cracknell
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28A. P. Cracknell and B. L. Davies, in Lecture Noles in Physics. edited by A. Janner, T. Boon, and M. Boon (Springer, Berlin, 1976), Vol. 50. R.Dirl 2080 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.88.53.18 On: Wed, 10 Dec 2014 09:16:26

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The Mathematical Theory of Symmetry in Solids

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yuppentheorie, Anwendungen in der Atom-wid Festkoyperpln'sik

- r Dirl
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17R. Dirl and P. Kasperkovitz, ('yuppentheorie, Anwendungen in der Atom-wid Festkoyperpln'sik (Vieweg, Braunschweig, 1977).