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In physics and mathematics, Wigner 3-jm symbols, also called 3j symbols,
are related to the Clebsch-Gordan coefficients of the groups SU(2) and SO(3) through
The 3j symbols show more symmetry in permutation of the labels than the corresponding Clebsch-Gordan coefficients.
The inverse relation can be found by noting that j1 - j2 - m3 is an integral number and making the substitution
The symmetry properties of 3j symbols are more convenient than those of
Clebsch-Gordan coefficients. A 3j symbol is invariant under an even
permutation of its columns:
An odd permutation of the columns gives a phase factor:
Changing the sign of the quantum numbers also gives a phase:
The Wigner 3j is zero unless
, is integer, and .
The contraction of the product of three rotational states with a 3j symbol,
is invariant under rotations.