New algorithms for the Molien function

Abstract

Two new forms are given for the Molien (generating) function for multiplicity cn1 of the identity representation in the symmetrized nth power of representation Γ of finite group G. These are M (Γ,G;z) =Σncn1zn=1/‖G‖Σg exp [Σlzlχ (gl)/l] =1/‖G‖ΣgΠj[1−zγj(g)]−δ j, where g is an element in G γj(g) an irreducible character in the Abelian subgroup A generated by g, δj the subduction coefficient of Γ of G↓γj of A; we usually take Γ irreducible. These forms have the merit of only requiring characters. In the following paper these algorithms are used to compute the Molien function for space group irreducible representation.