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 GP≡GH^'H(A, A^')ₙ conveys the following: the isomorphic crystallographic group G, a subgroup H^' of G, the largest normal subgroup H of G, contained in H^', and a transitive group of permutations P≡(A, A^')ₙ isomorphic to the factor group 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 DG^H^' of G associated with each GP and we reduce DG^H^' 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 $k=0$) 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.