Extended integrity bases of irreducible matrix groups-the crystal point groups

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Abstract
Any extended integrity basis (EIB) of a finite group can be composed from EIBs of irreducible matrix groups defined by irreducible representations of the group in question. The EIBs thus defined for irreducible matrix groups relevant to crystal point groups (and in virtue also to magneti point groups) are derived with the use of a previously reported algorithm based on the successive use of Clebsch-Gordan reductions. The EIBs of vector representation of crystal point groups are derived with the use of these fundamental EIBs. Division of invariants into denominator and numerator invariant enables one to find a general functional expression of an invariant or covariant with the use of EIBs. A possible approach to phase transition theory which goes beyond the usual Landau (polynomial) approximation is given to illustrate the use of EIBs.