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And thus give them the name. 2 project a tensor in V p ⊗ V¯ q onto a subspace λ. In practice one usually reduces a tensor step by step, decomposing a 2-particle state at each step. While there is some arbitrariness in the order in which these reductions are carried out, the final result is invariant and highly elegant: any group-theoretical invariant quantity can be expressed in terms of Wigner 3- and 6-j coefficients. 1 COUPLINGS AND RECOUPLINGS We denote the clebsches for μ ⊗ ν → λ by λ 11001100 00111100 μ 00111100 ν , Pλ = 111 000 000 111 000 111 000 111 000 111 000 111 μ 111 000 000 111 000 111 000 111 000 111 000 ν 111 λ111 000 000 111 000 111 .

0 . . λ2 ⎟ ⎜ ⎜ λ3 . . ⎟ ⎠ ⎝ 0 0 .. . . 6). In the matrix C(M − λ2 1)C † the eigenvalues corresponding to λ 2 are replaced by zeroes: ⎞ ⎛ λ1 − λ2 ⎟ λ1 − λ2 ⎜ ⎟ ⎜ λ1 − λ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ . ⎟, ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ λ3 − λ2 ⎟ ⎜ ⎟ ⎜ λ3 − λ2 ⎠ ⎝ .. and so on, so the product over all factors (M − λ 2 1)(M − λ3 1) . . , with exception of the (M − λ1 1) factor, has nonzero entries only in the subspace associated with λ1 : ⎞ ⎛ 1 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 1 ⎟ ⎜ ⎟ ⎜ † 0 C (M − λj 1)C = (λ1 − λj ) ⎜ ⎟.

50) and satisfy the completeness relation r Pi = 1 . 51) i=1 As tr(CPi C † ) = tr Pi , the dimension of the ith subspace is given by di = tr Pi . 48) that λi is the eigenvalue of M on P i subspace: MPi = λi Pi , (no sum on i) . 53) Hence, any matrix polynomial f (M) takes the scalar value f (λ i) on the Pi subspace f (M)Pi = f (λi )Pi . 54) This, of course, is the reason why one wants to work with irreducible reps: they reduce matrices and “operators” to pure numbers. 6 SPECTRAL DECOMPOSITION Suppose there exist several linearly independent invariant [d×d] hermitian matrices M1 , M2 , .