Matrix Products and the Explicit 3, 6, 9, and 12-j Coefficients of the Regular Representation of SU(n)

The explicit Wigner coefficients are determined for the direct product of regular representations, $\text{(N)⊗(N)=2(N)+…}$ , of SU(n), where N = n² − 1. Triple products C_mC_iC_m = αF_i + βD_i, and higher‐order products, are calculated, where C_i may be F_i or D_i, the N × N Hermitian matrices of the regular representation, and m is summed. The coefficients α, β are shown to be 6‐j symbols, and higher‐order products yield the explicit 9‐j, 12‐j, symbols. A theorem concerning (3p)‐j coefficients is proved.