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Ir ðG1 3 G2 Þ 5 Γi ðG1 Þ 3 Γj ðG2 Þ ð1:142Þ The direct product of two representations inside the same group is, in general, reducible Γred ðG1 3 G1 Þ 5 Γi ðG1 Þ 3 Γj ðG1 Þ ð1:143Þ 46 A Handbook of Magnetochemical Formulae which means that this is expressed as a sum of several IRs. 4 ð1:144Þ Littlewood’s Theorem On enlarging the subgroup SM 3 SN to the group SK (K 5 M 1 N) a question arises which IRs Γλ (SK) can be formed from IRs Γλ1 ðSM Þ and Γλ2 ðSN Þ: This process is formed through the outer product of IRs that decomposes as follows fΓλ1 ðSM Þ  Γλ2 ðSN Þg 5 X aðλ1 ; λ2 ; λÞU Γλ ðSK Þ ð1:145Þ λ The theorem can also be used to find the representations Γλ1 ðSM Þ 3 Γλ2 ðSN Þ into which Γλ(SK) decomposes on reduction to the subgroup SM 3 SN Γλ ðSK Þ 5 X aðλ; λ1 ; λ2 ÞU fΓλ1 ðSM Þ 3 Γλ2 ðSN Þg ð1:146Þ λ1 ;λ2 The theorem can be used to find which IRs UN½λŠ occur in the decomposition of the direct product UN½λ1 Š 3 UN½λ2 Š 5 X0 aðλ1 ; λ2 ; λÞU UN½λŠ ð1:147Þ λ where the summation is restricted only to those Young diagrams whose column lengths do not exceed N.

FΓ3 1 Γ5 g, T2 ! Γ5 S 5 0 ! Γ1 , Eg ! Γ3 S 5 1=2 ! Γ6 , Eg ! Γ3 S 5 3=2 ! Γ8 , Eg ! Γ3 S 5 2 ! fΓ3 1 Γ5 g, Eg ! Γ3 S 5 1 ! Γ4 , A2g ! Γ2 S 5 3=2 ! Γ8 , A2g ! Γ2 S 5 0 ! Γ1 , A1g ! Γ1 S 5 1=2 ! Γ6 , A1g ! Γ1 S 5 3=2 ! Γ8 , A1g ! Γ1 S 5 5=2 ! fΓ7 1 Γ8 g, A1g ! 19b 2 S 5 1=2 ! Γ6 , Eg ! Γ5 Γ6 3 Γ5 5 Γ6 1 Γ7 S 5 1 ! fΓ2 1 Γ5 g, Eg ! Γ5 fΓ2 1 Γ5 g 3 Γ5 5 fΓ5 g 1 fΓ1 1 Γ2 1 Γ3 1 Γ4 g S 5 3=2 ! fΓ6 1 Γ7 g, Eg ! Γ5 fΓ6 1 Γ7 g 3 Γ5 5 fΓ6 1 Γ7 g 1 fΓ6 1 Γ7 g S 5 2 ! fΓ1 1 Γ3 1 Γ4 1 Γ5 g, fΓ1 1 Γ3 1 Γ4 1 Γ5 g 3 Γ5 Eg !

For Un no diagram has more than n rows. 3. Making a journey through the diagram starting with the top row and entering each row from the right, at any point the number of ai’s encountered in any of the attached boxes must not exceed the number of previously encountered ai 2 1’s. 4. The numbers must not increase when reading across a row from left to right. 5. The numbers must decrease when reading a column from top to bottom. Molecular Symmetry 47 These rules ensure that states which were previously symmetrised are not antisymmetrised in the product and vice versa.

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