The representation group and its application to space groups

Abstract

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 -\. Three kinds of CSCO, the CSCO-I, -II, and -III, are defined which are the analogies of J 2 , (J 2 ,J Z ), and {J 2 ,J Z ,J Z )> respectively, for the rotation group S0 3 , where 7 2 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)

A complete list of symmetry adapted expressions to the fourth power for compact bending potentials in molecules with and symmetry from a general symbolic algebra program

“…For a discussion of this problem and alternative methods for the reduction of representations we refer to [16]. However, this algorithm offers a straightforward means to obtain a coherent set of reduced representation spaces and symmetry adapted coordinates.…”

Symmetries of symmetries and geometrical CP violation

“…Even though it is always possible to perform the according field redefinitions we think a comment is in order. Since it is, in principle, possible to distinguish the different components of a triplet, say H i and H j =i , from one another by appropriate measurements with respect to subgroups of (54), it is also possible to distinguish H i from a corresponding state H i = (U H ) i [22]. Therefore, the equivalence property of V can be used to reduce the size of the parameter space only if one does not insist on a relation between physical states and field operators to begin with.…”

Algebraic expressions for symmetry-adapted functions of the icosahedral group in spinor space

“…A convenient way of dealing with the d-v reps of a point or space group is by means of the so-called representation group (rep group) ®rst introduced by Chen et al (1985).…”

“…Several methods are available for ®nding d-v irreps of a group G; four of interest to us follow: (i) the doublegroup method (Bradley & Cracknell, 1972); (ii) the subduction method (McLellian, 1961); (iii) the representation group method where the problem of seeking the d-v irreps of G is equivalent to that of seeking the irreps of the rep group q of order jGj (Chen et al, 1985); and (iv) the projective representation method (Altmann, 1986). Obviously, the third approach is much simpler than the ®rst one, since the order q is one half of that of the group G y .…”