Clebsch-Gordan coefficients are the change-of-basis coefficients between an uncoupled tensor-product basis and a basis adapted to the irreducible decomposition of the tensor product.
For ,The identity follows by multiplying the weight characters and comparing their nested weight strings.
On the multiplicity-one summand , interchange of tensor factors acts by . A highest-weight vector exhibiting the sign is
The alternating summands in the tensor square give . Since is self-dual and has trivial determinant,
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The Clebsch–Gordan coefficients are numerical factors that arise in the study of angular momentum in quantum mechanics and in the theory of representations of groups, specifically the group \( SU(2) \) associated with rotations. They describe how to combine two angular momentum states into a total angular momentum state.