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 $L(\lambda) \downarrow^{S\_n}\_{S\_k \times S\_{n-k}}$?



Decompositions for Symmetric Groups
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*.


