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Im stuck with this one and I don't even know how to start, I would appreciate any help: Can you describe the induced representation of the standard representation of $S_{n}$ in $S_{n+1}$?

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Hint: do you know how to describe the standard representation as an induced representation? –  Qiaochu Yuan Jan 30 '12 at 2:15

1 Answer 1

Irreducible representations of $S_n$ are indexed by partitions of $n$. Assuming that by "standard representation" you mean the permutation representation, then this is the direct sum of the reps indexed by the partitions (n) and (n-1,1). Now there is a combinatorial rule for computing the induced representation: thinking of the partitions as corresponding to their Young diagrams, inducing the representation corresponding to a partition gives the sum over all partitions obtained by adding one box to the given partition. When you do this you get the direct sum

$$\mathrm{Ind}_{S_n}^{S_{n+1}} ((n) \oplus (n-1,1))=(n+1) \oplus (n,1) \oplus (n,1) \oplus (n-1,2) \oplus (n-1,1,1). $$

Edit: It's not clear from the way the problem is phrased what kind of description is sought. As Qiaochu indicates rather cryptically, another way to "describe" this representation would be to realize the permutation representation as the induction $\mathrm{Ind}_{S_{n-1}}^{S_{n}} ((n))$ and then use transitivity of induction to get the permutation representation of $S_{n+1}$ on the cosets of $S_{n-1}$ in $S_{n+1}$ (or, if you want, by the permutation action of $S_{n+1}$ on the set of ordered pairs of two integers $(i,j)$ with $1 \leq i \neq j \leq n+1$).

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