By Larry Davis

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**Example text**

12) ν By assumption λ, λ are irreps, so we have a recoupling relation between the exchanges in “s” and “t channels”: σ σ μ 00 11 ω 00 11 00 11 ρ ω μ ρ ν = dλ λ ν ρ σ λ σ λ μ λ ν μ . 13) ν We shall refer to as 3-j coefficients and as 6-j coefficients, and commit ourselves to no particular normalization convention. In atomic physics it is customary to absorb into the 3-vertex and define a 3-j symbol [238, 286, 345] ν 1 λ μ ν . 14) = (−1)ω λ α β γ μ λ μ ν Here α = 1, 2, . . , are indices, λ, μ, ν rep labels and ω the phase convention.

25) where D is a hermitian matrix with small elements, |D ab | on the conjugate space is given by 1. The action of g ∈ G (G† )b a = Ga b = δba − iDba . 26) D can be parametrized by N ≤ n 2 real parameters. N , the maximal number of independent parameters, is called the dimension of the group (also the dimension of the Lie algebra, or the dimension of the adjoint rep). In this monograph we shall consider only infinitesimal transformations of form 1. 11) connected to the identity. For example, we shall not consider invariances under coordinate reflections.

Bqq d d +δca11 (Ti )ac22 . . δcapp δbd11 . . δbqq + . . + δca11 δca22 . . (Ti )acpp δbd11 . . δbqq d d − δca11 δca22 . . δcapp (Ti )db11 . . δbqq − . . − δca11 δca22 . . δcapp δbd11 . . (Ti )bqq . 35) with a relative minus sign between lines flowing in opposite directions. The reader will recognize this as the Leibnitz rule. 60): (λ) C λ Ti Ti = Cλ λ 00111100 11001100 11001100 00001111 T 00111100 11001100 000 111 000 111 = λ 11 00 00 11 00 11 00 11 00 11 00 11 λ11 00 00 11 0 1 1 0 0 1 11 00 00 11 00 11 00 11 00 11 00 11 .