MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
This group is a direct product of $r$ wreath products of $S_i$ with $S_{m_i}$. Check out; I'm sure a lot is known about the representation theory of wreath products of symmetric groups. – user91132 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? – user91132 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
up vote 5 down vote accepted

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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.