The representation group and its application to space groups

A so-called representation (rep) group G is introduced which is formed by all the Vertical BarGVertical Bar 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 Jsup2, (Jsup2,Jsubz), and (Jsup2,Jsubz,J-barsubz), respectively, for the rotation group SOsub3, where J-barsubz 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 fatG in any subgroup symmetry adaptation can be found by solving the eigenequations of the CSCO-I, -II, and -III of fatG, respectively,more » in appropriate vector spaces.« less

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