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How do I take a tensor products of $SO(N)$ irreps and decompose it in terms of irreps for $N>3$? (I understand the special case of $SO(3)$ we can use the nice $SU(N)$ technology of Young Tableauxs but I am not interested in that case).

Are there general procedures? If not are there procedures that work for specific representations (fundamental, adjoint ,something else...)?

I already found this this very useful result in another answer (last part). This says that the fundamental representation always can be decomposed in terms of traceless, symmetric traceless and antisymmetric representations for all $SO(N)$. If we stick with just fundamental for a second what would happen when we take the tensor product with a third (and forth etc...) fundamental representation?

I am not that interested in $SO(6)$ results as it is basically similar to $SO(3)$ in that you can just use $SU(N)$ techniques. I am more interested in what you can do when there is no isomorphism to $SU(N)$.

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  • $\begingroup$ @ZeroTheHero, not really, I am interested in general $SO(N)$. I would like the most general results the most. If this is not possible, specific results can still be interesting but $SO(6)$ is not that interesting a specific case as I already understand that again you can use $SU(N)$ techniques due to the isomorphism of the lie algrebra to $su(N)$. I am more interesting in knowing what to do when such an isomorphism is not present. $\endgroup$
    – Kvothe
    Mar 16 '19 at 10:57
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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic ♦
    Mar 16 '19 at 11:08
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    $\begingroup$ @Qmechanic, perhaps. I considered it. However, there often is a language barrier when a physicist asks a group theory question at Mathematics. I would be fine with a move though if the powers that be consider that more appropriate. $\endgroup$
    – Kvothe
    Mar 16 '19 at 11:25
  • $\begingroup$ Related: physics.stackexchange.com/q/456932/196314. See my comment there as well. $\endgroup$
    – MannyC
    Mar 16 '19 at 13:29
  • $\begingroup$ It is also worth noting that if you are interested in having a program do this for you, then LieART (lieart.hepforge.org) can do exactly such calculations $\endgroup$
    – arow257
    Jul 29 '20 at 20:56
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The Mathematica application lieART can do decompositions for all classical and exceptional Lie algebras including $SO(N)$. The documentation in https://arxiv.org/pdf/1206.6379.pdf also explains the algorithm used to do the decomposition. The only downside is that it does not work with Young Tableaus for SO(N). This means that you will have to convert your irrep in terms of young tableau (or number of boxes per row, i.e. {4} for four index symmetric, or {1,1} for the anti-symmetric) into their Dynkin label description of the same irrep. Alternatively you can work with the description in terms of the dimensionality of the irrep but this has the downside that it is confusing when there are degeneracies in this dimension. I still found the latter easier so I converted all the young tableau to the dimension of the irrep in order to be able to feed it as input to lieART.

Some relevant literature can also be found in:

Group theory: Birdtracks, Lie's, and exceptional groups (paywalled)

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