Let $V=\mathbb{R}^n$ and $O(n)$ the orthogonal group acting with its standard action on $V$. Now for any partition $\lambda$ we have the Schur module $S_\lambda V$ which is a representation of $O(n)$. Unlike in the case of the general linear group, these are not irreducible as $O(n)$-modules (for example $\textrm{Sym}^2V$ decomposes to multiples of the identity matrix and the traceless symmetric matrices). Is there a general description/recipe on how the $O(n)$-module $S_\lambda V$ decomposes into irreducible ones? If not in general, maybe for some interesting (and not too special) types of partitions?
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1$\begingroup$ If you work over $\mathbb{C}$ instead of $\mathbb{R}$ there are branching rules due to Littlewood that give the multiplicities in terms of Littlewood-Richardson coefficients. I'm not sure in the real case though. $\endgroup$– NateCommented Aug 25, 2019 at 20:26
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$\begingroup$ That sounds like a good start. Do you have a good reference for these branching rules? $\endgroup$– HansCommented Aug 25, 2019 at 20:33
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$\begingroup$ I'm not sure which Littlewood paper they are from exactly, but they are at least stated in "Stable branching rules for classical symmetric pairs" by Howe, Tan, and Willenbring. $\endgroup$– NateCommented Aug 25, 2019 at 20:36
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$\begingroup$ Cool, thanks! I'll have a look. $\endgroup$– HansCommented Aug 25, 2019 at 20:39
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$\begingroup$ If you wrote this reference as an answer, I would be willing to accept it is an answer. $\endgroup$– HansCommented Aug 26, 2019 at 7:57
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