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Does anyone have a reference for a good description of representations of $S_{n}$ obtained by inducing up from $C_{S_{n}}(\pi)$, for some element $\pi$ of $S_{n}$? (I'd prefer an efficient combinatorial description if there is one.)

This seems like it should be doable, since I'm fairly sure that centralizers of elements are just going to be products (direct and/or wreath) of cyclic groups and smaller symmetric groups, so those representations should all be understandable combinatorially. Any references that people have would be great.

Edit: There are references for inducing from cyclic subgroups, as given in the answer to Decomposition of induced representations in S_n, and also Stembridge here: http://www.ams.org/mathscinet-getitem?mr=1023791, but I'm looking a bit more general than that.

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Induction from the long cycle to the whole symmetric group is famous and easy. Unfortunately induction from a wreath product of symmetric groups is called plethysm and has no known combinatorial rule in general. –  John Wiltshire-Gordon Jan 20 at 2:17
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Well, wreath products of symmetric groups with each other don't come up here. When forming the centralizer of a particular element $\pi$, you get direct sums of groups of the form $C_m \wr S_{k}$, where $\pi$'s cycle structure has $k$ cycles of length $m$. The full centralizer will be (I'm 95% sure) the direct sum of those over all $m$ that are present. –  coolpapa Jan 20 at 2:53

3 Answers 3

up vote 6 down vote accepted

I gave an answer for inducing the trivial representation in a comment at Decomposing the conjugacy representation of Sym$(n)$ for small $n$. It is in terms of a plethysm that seems to be just as intractable as the notorious $s_m[s_n]$.

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Thanks for that comment - do you know of answers to the question for other representations? (Even in terms of intractable plethysms.) –  coolpapa Jan 20 at 4:19
    
I've also thought through that comment and the exercise, and that sum: $$ \sum_{\lambda \vdash n} p_{\lambda} $$ will give you the character for the whole conjugation representation, which is the sum of the trivial representations induced up from many centralizers - one coming from each conjugacy class: $$ \sum_{K} \textrm{Ind}_{C(k)}^{S_{n}} 1_{C(k)} $$ So conceivably one of the elements of that sum might be easier to get a handle on. I'm not holding out much hope, though. –  coolpapa Jan 20 at 19:22

It is known from work of Roger Richardson and others that $G = S_{n}$ has a ``model", that is, we may choose one element $t_{i}$ from each conjugacy class of elements of order $2$, and a certain sign character $\lambda_{i}$ of each $C_{G}(t_{i}),$ such that each nontrivial irreducible character of $S_{n}$ occurs once and only once as an irreducible constituent of some ${\rm Ind}_{C_{G}(t_{i})}^{G}(\lambda_{i}).$ This is a more precise illustration of the fact that since all complex irreducible representations of the symmetric group are realizable over the real field ( even the rational field), we have $\sum_{ \chi \in {\rm Irr}(G)} \chi(1) = 1 +\sum_{i=1}^{k} [G:C_{G}(t_{i})]$ for the group $G = S_{n},$ (where $k = \lfloor \frac{n}{2} \rfloor $ is the number of conjugacy classes of elements of order $2$ of $S_{n}$) by a general formula obtained using the Frobenius-Schur indicator, which simplifies in the case of $S_{n}$ because we always have $\nu(\chi) = 1.$ Hence the sum of the induced characters at least has the right degree, but it is rather more subtle to obtain the finer result as stated.

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Thank you for the pointer. The relevant paper (ams.org/mathscinet-getitem?mr=1037615) has a couple of special cases of my question. I'd seen it referenced, but hadn't looked at it yet. –  coolpapa Jan 21 at 0:49

The representations $Ind_H^{G} \rho$, where $H=C_{G}(g)$ (for some $g\in G$) and $\rho$ is an irreducible representation of $C_{G}(g)$, all turn out to be irreducible representations of the quantum double of $G$ (denoted sometimes as $D(G)$). The action of the Hopf algebra $D(G)$ on this irreducible space is well known. For example, when reviewing this part in our paper, we describe the action of the Hopf algebra $D(S_n)$ on its irreducible representations. So if you want to know the action of $S_n$ on the induced representation $Ind_{C_{S_n}(\pi)}^{S_n}$, then you can use Eq. 3.9 by taking the trivial element of the dual group (i.e., replace $h^\ast$ by $\sum_{h\in G} h^\ast$). The rest of section 3 is specific to $S_n$. The centralizers are direct products of wreath products of the type $C_m \wr S_k$ as you mentioned. The irreducible representations of these groups can be obtained using Clifford theory. We also describe a coset factorization of $C_{S_n}(\pi)$ in $S_n$.

However, this does not answer how the induced representation decomposes into irreducible representations of $S_n$.

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