# What is an isoscalar factor?

I need to find a definition for "the isoscalar factors of 3j-symbols for the restriction $SO(n)\supset SO(n-1)$...denoted by brackets with a composite subscript $(n: n-1)$..." They are given as:

$$\left(\begin{array}{ccc}l_1&l_2&l_3\\l_1'&l_2'&l_3'\end{array}\right)_{\left(n: n-1\right)}$$

What does this mean? And can someone point me to an explicit definition? I'm familiar with the traditional Wigner 3j-symbols, my only inkling is that the Wigner 3j's are for SO(3) whereas

$$\left(\begin{array}{ccc}l_1&l_2&l_3\\l_1'&l_2'&l_3'\end{array}\right)_{\left(n\right)}$$

are for SO(n). But I have no idea what the composite subscript means, nor do I have a handle on what these coupling coefficients mean physically in SO(n).

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 This whole question looks strange because while $SO(3)$ irreducible representations are uniquely specified by a value of $L$, it is not the case for $SO(n)$ for $n>3$. For more complicated groups, one must use Young diagrams, not just a simple value of $L$, and the Racah coefficients - another name for the isoscalar factor - also depends on the whole diagram. Where did you see the comments and formulae you reproduced? – Luboš Motl Jan 21 '12 at 20:31 Its in eqs. 4.6-4.7 from "Coupling coefficients of SO(n)and integrals involving Jacobi and Gegenbauer polynomials" by Sigitas Alisauskas, you can find it at iopscience.iop.org/0305-4470/35/34/307 – okj Jan 21 '12 at 21:06 I see, so they're for the maximally symmetrized representations only. – Luboš Motl Jan 21 '12 at 21:10 Link also available in the preprint arXiv at arxiv.org/abs/math-ph/0201048 – Qmechanic♦ Mar 14 '12 at 13:47