# Coupling Coefficients in SO(4)

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:

$$\left(\begin{array}{ccc}l_1&l_2&l_3\\\left(l_1^',N_1\right)&\left(l_2^',N_2\right)&\left(l_3^',N_3\right)\end{array}\right)_{SO(4)}=\left(\begin{array}{ccc}l_1&l_2&l_3\\l_1^'&l_2^'&l_3^'\end{array}\right)_{\left(SO(4):SO(3)\right)}\left(\begin{array}{ccc}l_1^'&l_2^'&l_3^'\\N_1&N_2&N_3\end{array}\right)$$

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:

$$\left(\begin{array}{ccc}\left(X_1Y_1\right)&\left(X_2Y_x\right)&\left(XY\right)\\\left(M_{X_1}M_{Y_1}\right)&\left(M_{X_2}M_{Y_2}\right)&\left(M_XM_Y\right)\end{array}\right)_{SO(4)}=\left(\begin{array}{ccc}X_1&X_2&X\\M_{X_1}&M_{X_2}&M_X\end{array}\right)\left(\begin{array}{ccc}Y_1&Y_2&Y\\M_{Y_1}&M_{Y_2}&M_Y\end{array}\right)$$

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 $l_1$?)

$\bf SUPPLEMENTARY\ INFO$:

The first equation is eq. 4.6b from ftp://ftp.physics.uwa.edu.au/pub/Clebsch-Gordan/Papers/SO%28n%29.pdf

The second equation is eq. 22 from http://jmp.aip.org/resource/1/jmapaq/v51/i9/p093518_s1

The parameters of the second equation are defined as follows:

$L_{rs}\equiv-i(x_r\partial_s-x_s\partial_r)$

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

$X_k\equiv\frac{1}{2}(J_k+N_k)$
$Y_k\equiv\frac{1}{2}(J_k-N_k)$

$M_X=-X,...,X-1,X$
$M_Y=-Y,...,Y-1,Y$

$X=\left|X_1-X_2\right|,\left|X_1-X_2\right|+1,...,X_1+X_2$
$Y=\left|Y_1-Y_2\right|,\left|Y_1-Y_2\right|+1,...,Y_1+Y_2$

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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. arxiv.org/abs/1006.2875 –  Qmechanic Mar 14 '12 at 13:17