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Symmetry group 0j symbol

  1. Mar 28, 2013 #1
    Hello everyone,

    I read in Edmond's 'Angular momentum in Quantum Mechanics' that the symmetry group of the 9j symbol is isomorphic to the group [itex]S_3 \times S_3 \times S_2[/itex].

    Why is this? Can anyone shed some light on this?
  2. jcsd
  3. Mar 28, 2013 #2


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    Edmonds says,
    Likewise, the Wikipedia page on 9-j symbols describes it this way.
  4. Mar 28, 2013 #3
    I don't see the full picture, yet.

    Labelling the rows [itex]r_1,r_2,r_3[/itex] and the columns [itex]c_1,c_2,c_3[/itex], it's easy to show that the subgroups of the row and column operations are both isomorphic to [itex]S_3[/itex]. Since any row permutation does not affect the order of the [itex]c_i[/itex], its an element of [itex]S_3 \times e[/itex] and any column permutation is in [itex]e \times S_3[/itex] in the same way.

    So the subgroup of all symmetry operations not containing a transposition of the array is [itex]S_3 \times S_3[/itex]. But how do you take the transpositions into account?

    I see that relevant subgroup is [itex]S_2[/itex], but I don't see exactly how you go from [itex]S_3 \times S_3[/itex] to [itex]S_3 \times S_3 \times S_2[/itex] by taking the transpositions into account.
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