Let C be a cyclic subgroup of S_n. How does the representation $Ind_C^{S_n}\rho$, where $\rho$ is some representation of $C$, decompose into irreducible components? Is there are a way to know which components appear with multiplicity 1?
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4$\begingroup$ There are two formulas in Symmetric Functions and Hall Polynomials that might be useful, which give you the character of the induced representation written in the basis of power sums. The case where C is generated by an n-cycle and the representation is faithful appears as Exercise 7.12, and the case of C an arbitrary subgroup and the representation is trivial is Exercise 7.4. $\endgroup$– Dan PetersenFeb 8, 2011 at 16:47
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2$\begingroup$ If you just care about multiplicities, you might want to consider restrictions instead of inductions (Frobenius equivalence). $\endgroup$– darij grinbergFeb 8, 2011 at 17:29
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1 Answer
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There is a combinatorial way to decompose $Res_C^{S_n}S^\lambda$ for an irreducible $S_n$-module $S^\lambda$. We use the notion of the "major index" for a standard tabuleau of shape $\lambda$. If $C=C_n$, the result is obtained by Kraśkiewiz-Weyman. See also Garsia's paper "Combinatorics of the free Lie algebra and the symmetric group"(Theorem 8.4) and Reutenauer's book "Free Lie algebras"(Theorem 8.8 and 8.9). This result can be generalized to any cyclic subgroup in Jöllenbeck-Schocker's paper "Cyclic characters of symmetric groups".
