Physical applications of crystallographic color groups: Landau theory of phase transitions

The simplest crystallographic color groups are the permutational color groups. Elements of these groups combine two types of transformations: One is a rotation and/or translation of physical space and the other is a permutation. The groups considered here are subgroups of direct products and abstractly isomorphic to crystallographic groups, hence their relative simplicity. Despite this simplicity, there is a richness of information contained in each such group. The group symbol $G^P\equiv \frac{\frac{G}{H^{\prime }}}{H{(A, A^{\prime })}_n}$ conveys the following: the isomorphic crystallographic group $G$, a subgroup $H^{\prime }$ of $G$, the largest normal subgroup $H$ of $G$, contained in $H^{\prime }$, and a transitive group of permutations $P\equiv {(A, A^{\prime })}_n$ isomorphic to the factor group $\frac{G}{H}$. We derive and tabulate here all classes of equivalent permutational color point groups using a definition of equivalence classes which we physically motivate. For applications we require and report here the permutation representation $D_G^{H^{\prime }}$ of $G$ associated with each $G^P$ and we reduce $D_G^{H^{\prime }}$ into irreducible components. The major application given here is to the Landau theory of symmetry change in continuous phase transitions. A complete set of tables is presented for all allowed equitranslational ("Zellengleich" or $k=0\mathrm{}$) phase transitions in crystals based on group-theoretical criteria, including a new "kernel-core" criterion. The tables may be used to determine all active representations for transitions between two specific groups or alternatively, all possible subgroups which can be obtained from a specific group and irreducible representation. We also relate two classifications schemes for phase transitions to the structure of permutational color groups.