The Wigner 
-symbols
are a generalization of Clebsch-Gordan
coefficients and Wigner 3j- and 6j-symbols which arises in the coupling of
four angular momenta. They can be written in terms of the Wigner
3j- and Wigner 6j-symbols.
Let tensor operators 
and 
act, respectively, on subsystems 1 and 2. Then the reduced matrix element of the
product 
of these two irreducible operators in the coupled representation is given in terms
of the reduced matrix elements of the individual operators in the uncoupled representation
by
![(tau^'tau_1^'j_1^'tau_2^'j_2^'J^'∥[T^((k_1))×U^((k_2))]^((k))∥tautau_1j_1tau_2j_2J)
=sqrt((2J+1)(2J^'+1)(2k+1))sum_(tau^('')){j_1^' j_1 k_1; j_2^' j_2 k_2; J^' J k}
×(tau^'tau_1^'j_1^'∥T^((k_1))∥tau^('')tau_1j_1)(tau^('')tau_2^'j_2^'∥U^((k_2))∥tautau_2j_2),](/images/equations/Wigner9j-Symbol/NumberedEquation1.svg) |
(1)
|
where 
is a Wigner 
-symbol
(Gordy and Cook 1984).
In terms of the 
-symbols,
 |
(2)
|
(Messiah 1962, p. 1067; Shore and Menzel 1968, pp. 282-283).
In terms of the 
-symbols,
 |
(3)
|
(Messiah 1962, p. 1067; Shore and Menzel 1968, p. 282).
A 
-symbol
is invariant under reflection through one of the diagonals, and becomes multiplied
by 
upon the exchange of two rows or columns, where 
(Messiah 1962, p. 1067). It also satisfies
the orthogonality relationship
 |
(4)
|
(Messiah 1962, p. 1067).
Explicit formulas include
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
where
 |
(9)
|
(Messiah 1962, p. 1068; Shore and Menzel 1968, p. 282).
'
Symbols." Appendix C.III in