►The quantities in the symbol are called angular momenta.
…The corresponding projective quantum numbers
are given by
…
►where is defined as in §16.2.
►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
…
►An often used alternative to the symbol is the Clebsch–Gordan coefficient
…For other notations for , , symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).
…
►Thus is the permutation , , .
…
►If, for example, a permutation of the integers 1 through 6 is denoted by , then the cycles are , , and .
…The function also interchanges 3 and 6, and sends 4 to itself.
…
►As an example, , , is a partition of .
…
►
…
►When any one of is equal to , or , the symbol has a simple algebraic form.
…For these and other results, and also cases in which any one of is or , see Edmonds (1974, pp. 125–127).
…
►
§34.3(ii) Symmetry
►Even permutations of columns of a symbol leave it unchanged; odd permutations of columns produce a phase factor , for example,
…
►The symbol is defined by the following double sum of products of symbols:
…
►Except in degenerate cases the combination of the triangle inequalities for the four symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths ; see Figure 34.4.1.
…
►where is defined as in §16.2.
►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
►In a symbol, if the three angular momenta do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the symbol is zero.
Similarly the symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four symbols in the summation.
…However, the and symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled.
…