The representation group and its application to space groups

A so-called representation (rep) group G is introduced which is formed by all the $|\mathrm{G}|$ distinct operators (or matrices) of an abstract group \ifmmode \hat{G}\else \^{G}\fi{} in a rep space $L$ and which is an $m$-fold covering group of another abstract group g. G forms a rep of \ifmmode \hat{G}\else \^{G}\fi{}. 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 ${J}^{2}$, (${J}^{2}$,${J}_{z}$), and (${J}^{2}$,${J}_{z}$,${\overline{J}}_{z}$), respectively, for the rotation group S${\mathrm{O}}_{3}$, where ${\overline{J}}_{z}$ 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 \ifmmode \hat{G}\else \^{G}\fi{} in the space $L$. Therefore, the construction of the irreps of \ifmmode \hat{G}\else \^{G}\fi{} 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 $|\mathrm{G}|$ is no more than that for the group g with order $|\mathrm{g}|=\frac{|\mathrm{G}|}{m}$, and thus tremendous labor can be saved by working with the rep group G instead of the abstract group \ifmmode \hat{G}\else \^{G}\fi{}. 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 ${\mathbf{G}\ensuremath{'}}_{\mathrm{k}}$. The $n$ inequivalent irreps of ${\mathbf{G}\ensuremath{'}}_{\mathrm{k}}$ 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 ${\mathbf{G}}_{0}$(k). An easily programmable algorithm is established for computing the Clebsch-Gordan series and Clebsch-Gordan coefficients of a space group simultaneously.