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 jm1, m2 = $$\langle$$ j1 j2 m1 m2 | j1 j2, j m$$\rangle$$, with

|j, m$$\rangle$$ = ∑m1+m2=m $$\langle$$ j1 j2 m1 m2 | j1 j2, j m$$\rangle$$ |m1 m2$$\rangle$$ ,   j1 + j2 = j.

* Properties:

$$\langle$$ j1 m1, j2 m2 | j m$$\rangle$$ = (–1) j1+j2j $$\langle$$ j2 m2, j1 m1 | j m $$\rangle$$ .

* Relationships: In terms of 6j-symbols,

$\langle\, j_{_1}\,j_{_2}m_{_1}m_{_2}\vert\, j_{_1}j_{_2},j\,m\, \rangle = (-1)^{-j_1+j_2-m}\,\sqrt{\vphantom{\sqrt1}2j+1} \left(\matrix{j_{_1} & j_{_2} & j \cr m_{_1} & m_{_2} & -m}\right) .$

* Theorem: πj ⊗ πj' = πj+j' ⊕ πj+j'–1 ⊕ ··· ⊕ π| j–j' |.

References
@ Calculation: Klink & Wickramasekara EJP(10) [simple method]; Horst & Reuter CPC(11)-a1011 [CleGo computer package]; Ibort et al a1610 [numerical algorithm].
@ 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 [Uq(su(3))]; Wu mp/06 [permutation group].