# 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).

• 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? Jan 21, 2012 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, 2012 at 21:06
• I see, so they're for the maximally symmetrized representations only. Jan 21, 2012 at 21:10
• Link also available in the preprint arXiv at arxiv.org/abs/math-ph/0201048 Mar 14, 2012 at 13:47

When labelling states using a subgroup chain like SO(n)$\to$SO(n-1), it is also possible to express the Clebsch-Gordan coefficient as a product of an SO(n-1) Clebsch multiplied by an "isoscalar factor", which is the part of the SO(n) Clebsch not in SO(n-1).
In the specific case of SO(3), the subgroup usually used is SO(2): the subgroup of rotations about the z-axis. This subgroup is Abelian so the representations are 1-dimensional, labelled by M, and just exponentials $e^{iM \varphi}$ for instance. For SO(2) the Clebsch are thus 1 if the projections satisfy the correct addition rule $M_1+M_2=M_3$. Thus, in this case, the isoscalar factor for SO(3) is the Clebsch itself since the SO(2) Clebsch is just 1 (or more correctly a delta function enforcing the condition $M_1+M_2=M_3$.)
• To clarify, each irreducible representation of $SO(n)$ contains at most one copy of any given $SO(n-1)$ irrep. However, the tensor products can have multiplicities and the extra labels are needed for those. Aug 1, 2017 at 20:22