# Representations of semidirect products of symmetric groups

This is sort of a vague (I apologize in advance) question, but I'm interested in the representation theory of the following group

$A \rtimes B$, where $A = (S_1)^{m_1} \times (S_2)^{m_2} \times \ldots \times (S_r)^{m_r}$, $B = S_{m_1} \times S_{m_2} \times ... \times S_{m_r}$, and $B$ acts on $A$ by permuting the factors. Is something nice known about the representation theory of these groups? Does anyone know a good reference for something like this?

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This group is a direct product of $r$ wreath products of $S_i$ with $S_{m_i}$. Check out en.wikipedia.org/wiki/Wreath_product; I'm sure a lot is known about the representation theory of wreath products of symmetric groups. – Konstantin Ardakov Jan 5 '11 at 20:02
This is a direct product of permutational wreath products of $S_i$ and $S_{m_i}$. Right? So you need to know the representation theory of one of the direct factors $S_i \wr S_{m_i}$ (I assume the field is complex numbers). I do not know the representation theory of one wreath product, but it must be known. – Mark Sapir Jan 5 '11 at 20:06
@Konstantin: This is not a "usual" wreath product, it is a permutational wreath product (the top group is a permutation group on a set $S$, and the bottom group is a direct product of $|S|$ copies of some group). – Mark Sapir Jan 5 '11 at 20:08
@Mark: I'm not aware of any distinction between "usual" and "permutational" wreath products; what in your language is a "usual" wreath product? – Konstantin Ardakov Jan 5 '11 at 20:22
Macdonald has a nice treatment of this in one of the appendices of Symmetric Functions and Hall Polynomials. – Andy B Jan 5 '11 at 20:34

## 1 Answer

The representations of wreath products of symmetric groups are known: for example, see section 4.3 of "The representation theory of the symmetric group" by James and Kerber.

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Hey, I'm interested in similar questions. I've checked the book your recommended...Do you know any good reference on the representation ring structure of wreath products? Especially the representation theory for the product G\wr S_n where G is not finite. Thanks! – Megan Aug 22 '12 at 0:02