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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.

Inverse relation

The inverse relation can be found by noting that j1 - j2 - m3 is an integral number and making the substitution

Symmetry properties

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:

Selection rules

The Wigner 3j is zero unless , is integer, and .

Scalar invariant

The contraction of the product of three rotational states with a 3j symbol,

is invariant under rotations.

Orthogonality Relations