I'm looking for information about how representations of $S_n$ decompose under restriction.

I know about the branching rule: That is, in characteristic 0, irreducible modules $L(\lambda)$ for $S_n$ decompose into a direct sum of irreducible modules of $S_{n-1}$ and there is a nice combinatorial rule for determining the multiplicities of the $S_{n-1}$-modules in the decomposition.

Are there similar results for the embedding of $S_k \times S_{n-k}$ in $S_n$? That is, is there a way of computing the multiplicity of the irreducible $S_k \times S_{n-k}$-module $L(\nu) \otimes L(\mu)$ in $L(\lambda) \downarrow^{S_n}_{S_k \times S_{n-k}}$?

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    $\begingroup$ By Frobenius reciprocity, you're just looking for the Littlewood-Richardson rule. $\endgroup$ – Alexander Woo Oct 5 '10 at 20:58
  • $\begingroup$ The answer is almost certainly yes, but I'd have to do some looking for the most appropriate references. The subgroups you mention are among the Young subgroups; inducing or restricting characters from or to them is a classical theme in the subject, going back I presume to Young himself. $\endgroup$ – Jim Humphreys Oct 5 '10 at 21:01

To elaborate on Alexander Woo's comment, the multiplicity of $L(\nu) \otimes L(\mu)$ in $L(\lambda)$ restricted is the Littlewood-Richardson coefficient $c^\lambda_{\nu, \mu}$. See http://en.wikipedia.org/wiki/Littlewood-Richardson_coefficient for the statement. For a proof (i.e., why this is related to symmetric functions or representations of the general linear group), you can see Section 7.18 of Stanley's Enumerative Combinatorics vol. 2.


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