Clebsch-Gordan Theory / Coefficients

In General > s.a. angular momentum; group representations; SU(2) [6j, 10j symbols].
* Idea: Mathematical symbols used to integrate products of spherical harmonics, or add angular momenta in quantum mechanics.
$ Def: The coefficients C ^j_{m_1, m_2} = j₁ j₂ m₁ m₂ | j₁ j₂, jm, with

|j, m = _{m_1+m_2=m} j₁ j₂ m₁ m₂ | j₁ j₂, j m |m₁ m₂ ,   j₁ + j₂ = j.

* Properties:

j₁ m₁, j₂ m₂ | j m = (–1) ^j_1+j_2–j j₂ m₂, j₁ m₁ | j m .

* Relationships: In terms of 6j-symbols,

* Theorem: _{j j'} = _{j+j' j+j'–1} ··· _{| j–j' |}.

References
@ Asymptotics: de Guise & Rowe JMP(98), Reinsch & Morehead JMP(99).
@ For SU(3): Coleman JMP(64); Grigorescu SCF(84)mp/00; Prakash & Sharatchandra JMP(96), Rowe & Repka JMP(97) [and SU(2)]; Rowe & Bahri JMP(00).
@ Other groups: Asherova et al PAN(01)m.QA [U_q(su(3))]; Wu mp/06 [permutation group].

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