C.1 Clebsch–Gordon expansions

1 0 i 1 1 1 Y1iY1j = √--𝜖ijkY1k + -Y2ij, 2 √ -- 2 1 −1 1- i--2 0 -1- 0 Y1iY1j = 3δij − 4 𝜖ijkY1k − 12 Y2ij, 2 1 Y10iY10j = -δij + -Y20ij 3 3
Y 11iY22ij = Y 33ijk, ( ) Y 0Y0 = − 4-δkjY0 + 6- δijY 0 + δikY 0 + 1Y 0 , 1i 2jk 5 1i 5 1k 1j 5 3ijk 1 0 2- 1 3- 1 3- 1 -i--( 1 1) 2- 1 Y1iY2jk = 5Y1iδjk − 5Y1jδik − 5 Y1kδij + √2-- 𝜖iklY2jl + 𝜖ijlY2kl + 5 Y3ijk, ( ) Y11iY12jk = − 1-∂ Y11iY 02jk , 6 √ -- −1 1 3 0 3 0 1 0 i 2 ( 0 0) 1 0 Y 2ij Y 1k = 10Y1iδjk + 10Y1jδik − 5Y1kδij +-12- 𝜖jklY2il + 𝜖iklY2lj − 30-Y3ijk, 2 3 3 i ( ) 4 Y10iY12jk = − --Y11iδjk + --Y11jδik + -Y 11kδij − -√--- 𝜖iklY12jl + 𝜖ijlY12kl + ---Y31ijk, 5 5 5 3 2- 15 3 3 1 i√ 2 ( ) 1 Y22ijY−1k1= --Y10iδjk + --Y10jδik − -Y10kδij −---- 𝜖jklY20il + 𝜖iklY20lj −---Y30ijk, 2 0 10( 2 − 1)10 5 12 30 Y2ijY 1k = ∂ Y2ijY1k

The Clebsch–Gordon expansions involving two l = 2 harmonics have been used in the text. They are fairly long and are not given here but can be found in [59].


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