Clebsch-Gordan coefficients of discrete groups in subgroup bases

TL;DR

This paper introduces a method to express Clebsch-Gordan coefficients of discrete groups using subgroup coefficients and embedding factors, simplifying calculations and clarifying phase ambiguities.

Contribution

It provides a systematic approach to decompose CG coefficients into subgroup components and defines embedding factors with fixed phases, enhancing computational efficiency.

Findings

Derived embedding factors for PSL_2(7) using subgroups S_4 and T_7.

Reduced phase ambiguities to sign ambiguities in certain cases.

Established invariance of embedding factors under basis transformations.

Abstract

We express each Clebsch-Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase ambiguities. Particularly, they are invariant under basis transformations of irreducible representations of both the group and its subgroup. We then impose on the embedding factors constraints, which relate them to their counterparts under complex conjugate and therefore restrict the phases of embedding factors. In some cases, the phase ambiguities are reduced to sign ambiguities. We describe the procedure of obtaining embedding factors and then calculate CG coefficients of the group \mathcal{PSL}_{2}\left(7\right) in terms of embedding factors of its subgroups S_{4} and \mathcal{T}_{7}.