Symmetry group 0j symbol

  • Thread starter Yoran91
  • Start date
  • #1
37
0
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?
 

Answers and Replies

  • #2
Bill_K
Science Advisor
Insights Author
4,155
195
Edmonds says,
we may permute the rows or columns in the matrix forming the 9-j symbol, or transpose the matrix itself...The symmetry group [is] the product of the three permutation groups of three, three and two objects respectively.
Likewise, the Wikipedia page on 9-j symbols describes it this way.
 
  • #3
37
0
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.
 

Related Threads on Symmetry group 0j symbol

Replies
4
Views
1K
  • Last Post
Replies
3
Views
3K
Replies
4
Views
677
Replies
29
Views
5K
Replies
5
Views
5K
Replies
5
Views
2K
Replies
11
Views
3K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
4
Views
2K
Top