Monomial representation of point-group symmetries

Abstract

Most irreducible-matrix representations in point-group symmetries can adopt monomial form. In that case all representational matrices have only one nonzero element in each row and column. The so-obtained standard basis choice contrasts with the conventional Wigner-Racah option. Monomial representations give rise to interesting properties of the corresponding Clebsch-Gordan series: All coupling coefficients are equal in absolute value and a natural intrinsic multiplicity separation is obtained. The concept is also useful in reaching a consistent solution of the multiplicity problem in the reduction of direct products, involving the fourfold ${U}^{\ensuremath{'}}$ representation of the octahedral spinor group. Several tables of basis transformations and coupling coefficients in octahedral and icosahedral symmetries are included.