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I have two equations (from two distinct authors) for the decomposition of a coupling coefficient of SO(4) (i.e. Wigner 3j-symbol for SO(4)). In the first:


The left hand side is the coupling (Wigner) coefficient for SO(4) and the right hand side has an isoscalar factor with the label $SO(4):SO(3)$, and a normal Wigner coefficient for SO(3).

In the second equation the author factors the SO(4) coupling coefficient into the product of two SO(3) coupling coefficients as:


In this case SO(4) is from the direct product of two SO(3)s: $(X_1Y_1)\bigotimes(X_2Y_2)\rightarrow(XY)$

$\bf QUESTION$: I want to set these equations equal to each other and solve for the isoscalar factor, but I am confused by the fact that the first author only uses a scalar for the upper arguments whereas the second author uses a tuple. How do the parameters of the SO(4) coupling coefficients equate? (e.g. Is there a way to get $X_1,Y_1$ from $l_1$?)


The first equation is eq. 4.6b from

The second equation is eq. 22 from

The parameters of the second equation are defined as follows:


$J_r\equiv\frac{1}{2}\varepsilon_{rst}L_{st}$, $N_r\equiv L_{r4}$, i.e. $$\begin{array}{ccc}J_1=L_{23}&J_2=L_{31}&J_3=L_{12}\\N_1=L_{14}&N_2=L_{24}&N_3=L_{34}\end{array}$$




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I have noticed one thing that might help, the author of eq. 1 states that "The canonical basis states of the symmetric (class-one) irreducible representation $l=l_{(n)}$ for the chain $SO(n)\supset SO(n-1)\supset...\supset SO(3)\supset SO(2)$ are labelled by the $(n-2)$-tuple $M=(l_{(n-1)},N)=(l_{(n-1)},...,l_{(3)},m_{(2)})$ of integers $l_{(n)}\geq l_{(n-1)}\geq ...\geq l_{(3)}\geq |m_{(2)}|$". Also in his notation here I believe "SO(n) irreducible representation $l_{(n)}=l$ [has] SO(n-1) irrep labels $l_{(n-1)}=l'$". – okj Jan 24 '12 at 19:32
The first ftp link does not work for me. The second link is behind a pay-wall. In the future, please link to an arXiv abstract page if possible, e.g. – Qmechanic Mar 14 '12 at 13:17

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