Calculation of the Molien generating function for invariants of space groups

Abstract

The new algorithms for computing the Molien generating function for a representation of a finite group obtained in the preceding paper are applied to obtain an expression which can be used for irreducible representation (*kn) of any crystallographic space group G. It proves convenient to express M (Γ,G;z) as a sum: M (Γ,G;z) =1/‖P‖Σkckm̄ (Γ,gk;z), where the partial Molien function m̄ (Γ;gk;z) is labelled by a coset representative gk carrying the class index k of the point group P=G/T, T being the translation group, the sum is over classes k, and ck is the order of class Ck(P). The resulting form was used to compute M (Γ,G;z) for irreducible representations *Γn, *Xn, *Rn of nonsymmorphic space group A-15 or O3h-Pm3n in which many high temperature superconducting crystals occur. Certain of these representations (matrix groups) are identified as generalized Coxeter groups, i.e., unitary groups generated by reflections. The Molien function for these groups as the required form given by Shephard and Todd: M (Γ,G;z) =[Πi(1−zdi)]−1. The di satisfy dimensionality theorems.