Angular Momentum Vector Coupling Algebra

Clebsch-Gordan coefficient

C ^JM_{j1 m1, j2 m2}  often also denoted as <j1 m1, j2 m2|j1 j2 J M> and referred to as vector-coupling or Wigner coefficient. The calculation uses the phase convention [1,2], where <j1 j1, j2 m2|j1 j2 J J> is positive for all allowed J (m2=J-j1)  

Three-j Symbol

The Wigner 3-j symbol is a more symmetric expression of the vector coupling, directly related to the Clebsch-Gordan coefficient. For symmetry property and other relations see [1].

Six-j Symbol

The Wigner 6-j symbol also known as Racah 6-j symbol, expresses a re-coupling of three angular momenta.  Given j1, j2, j3 the total momentum J can be found as j12+j3 or via j1+j23, where j12=j1+j2 and j23=j2+j3. The Wigner 6-j symbols give the relation between these kind of coupling schemes. The symbol is invariant under any permutation of columns and is invariant against interchange of the lower and upper arguments in each of any two columns. Precise definitions and relations can be found in [1]. 

If you have question/comments fell free to drop me a line Alexander Volya

References:

[1] A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press 1974.

[2] E. Condon, and G. Shortley, The Theory of Atomic Spectra, Cambridge, 1935.