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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 $L(\lambda) \downarrow^{S_n}_{S_k \times S_{n-k}}$?

flag	asked Oct 5 2010 at 20:29 Beth 33●4

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By Frobenius reciprocity, you're just looking for the Littlewood-Richardson rule. – Alexander Woo Oct 5 2010 at 20:58

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. – Jim Humphreys Oct 5 2010 at 21:01

Steven Sam · Accepted Answer · 2010-10-05 21:18:43Z UTC

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.

Decompositions for Symmetric Groups

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