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The branching laws for the $SO(n-1)$ as a subgroup of $SO(n)$ are well known and easy to find. See for example the Wikipedia article:

https://en.wikipedia.org/wiki/Restricted_representation#CITEREFMurnaghan1938

I am having trouble finding the next most complicated example, i.e. $$ SO(n-2) \times SO(2) \subseteq SO(n). $$ Where can a description of these branching laws be found?

Edit: In particular I would like to know examples of representations of $SO(n-2) \times SO(2)$ which appear with multiplicity $1$ in any branching.

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    $\begingroup$ For $\mathrm{O}(n-2)\times\mathrm{O}(2)\subseteq\mathrm{O}(n)$, see Proposition 10.2 of "Young Tableaux, Gelfand Patterns, and Branching Rules for Classical Groups" by Robert A. Proctor. $\endgroup$
    – Peter Humphries
    Commented Apr 12, 2018 at 0:23
  • $\begingroup$ @Peter: Thank you a lot for reference! It is somewhat technical for me unfortunately. Can we see from Proctor's results, that for example, when the branchings are multiplicity free? I.e. are branching always multiplicity free, or maybe just for the fundamental representations? $\endgroup$
    – Esra Sümeyye
    Commented Apr 12, 2018 at 11:13
  • $\begingroup$ check Eastwood-Wolf, branchig of ...., Arxive 0812.0822 math[RT] in this paper you find who to compute branching laws useing LiE. $\endgroup$
    – jorge vargas
    Commented Apr 18, 2018 at 15:48
  • $\begingroup$ This does seem to get very complicated as you go deeper into the subgroups. By the way, the arXiv link cited by Jorge Vargas is here: front.math.ucdavis.edu/0812.0822 (and the paper itself was published in 2011 in the Munster Math. J., probably not different from the arXiv preprint). Also, the longer paper by Proctor is freely available online here: www-sciencedirect-com.silk.library.umass.edu/science/article/… $\endgroup$
    – Jim Humphreys
    Commented Apr 18, 2018 at 16:12
  • $\begingroup$ Incidentally, your subgroup is a Levi subgroup, so branching can be computed using crystals. That would take some work and not get you results as precise as what's already mentioned. $\endgroup$
    – Allen Knutson
    Commented May 6, 2018 at 13:24

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It is not multiplicity free..., check the green book of Antony Knapp, or else the old book of Zelobenko, compact.....

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  • $\begingroup$ Ok, so not multiplicity free in general. $\endgroup$
    – Esra Sümeyye
    Commented Apr 13, 2018 at 16:56
  • $\begingroup$ However, since we have a Gelfand-pair, the trivial representation of $SO(n-2) \times SO(2)$ can appear only once in the decomposition of any $SO(n)$-module. $\endgroup$
    – Esra Sümeyye
    Commented Apr 13, 2018 at 16:57
  • $\begingroup$ Is it naive to assume that such a statement holds true for the fundamental representation of $SO(n-2) \times SO(2)$? Or even for its exterior powers? $\endgroup$
    – Esra Sümeyye
    Commented Apr 13, 2018 at 16:59
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check Eastwood-Wolf, branchig of ...., Arxive 0812.0822 math[RT] in this paper you find who to compute branching laws useing LiE.

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