Clebsch-Gordan coefficients of discrete groups in subgroup bases

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}.

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References in corpus (6)

• Tri-Bimaximal Neutrino Mixing and the Family Symmetry Z_7 x Z_3
• Lepton Mixing and Cancellation of the Dirac Mass Hierarchy in SO(10) GUTs with Flavor Symmetries T7 and Sigma(81)
• Simple Finite Non-Abelian Flavor Groups
• Observable T_7 Lepton Flavor Symmetry at the Large Hadron Collider
• Neutrino masses, the $\mu$-term and ${\mathcal{P}\mathcal{SL}}_2(7)$
• PSL(2,7) Representations and their relevance to Neutrino Physics