# 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?

• 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. 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.
– user6976
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).
– user6976
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? Jan 5 '11 at 20:22
• Macdonald has a nice treatment of this in one of the appendices of Symmetric Functions and Hall Polynomials. Jan 5 '11 at 20:34