## Reference request: representation theory of the hyperoctahedral group

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I was wondering if someone knows a good reference for the representation theory of the hyper-octahedral group $G$. The hyper-octahedral group $G$ is defined as the wreath product of $C_2$ (cyclic group order $2$) with $S_n$ (symmetric group on $n$ letters).

I understand that the representations of $G$ are in bijection with bi-partitions of $n$. I am looking for a reference which explains the details of why the representations of $G$ are in bijection with bi-partitions of $n$, and constructs the irreducible representations of $G$ (I imagine this is vaguely similar to the construction of Specht modules for $S_n$).

So far, the only reference I have is an Appendix of MacDonald's "Symmetric functions and Hall polynomials" (2nd version), which deals with the representation theory of the wreath product of $H$ with $S_n$ (for $H$ being an arbitrary group, not $C_2$).

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3
"MacDonald" here (and often elsewhere) refers to Ian G. Macdonald, whose work has been highly influential in representation theory, combinatorics... His contemporary Ian D. MacDonald (sometimes I.D. Macdonald) had a quirky professional career and did much less influential work in conventional group theory. Anyway, I.G. Macdonald wrote an interesting note Some irreducible representations of Weyl groups (Bull. LMS, 1972) with reference to W. Specht's 1937 paper Darstellungstheorie der Hyperoktaedergruppe. But more recent references are suggested in the answers here. – Jim Humphreys May 23 2010 at 13:20

## 3 Answers

I liked the references of Kerber listed in the wikipedia article. The most relevant chapter is available online, along with both volumes which were quite useful.

Kerber's presentation focusses on the idea that H is going to be cyclic and specifically handles H of order 2, but like MacDonald handles general H abstractly. GAP handles the hyper-octahedral group this way too, using generic code for wreath products written more or less solely for the hyper-octahedral group. The "bi" in bi-partitions just refers to the two conjugacy classes of C2, and the general theory replaces "bi" by however many conjugacy classes H has.

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The theory extends to the wreath product of a cyclic group $C$ with the symmetric group $S_n$. Then we are looking at a list of $|C|$ partitions with a total of $n$ boxes. This is in MacDonald so I expect you know this.

Now we can deform these group algebras analogously to deforming the group algebras of $S_n$ to Hecke algebras. The group algebra of the cyclic groups is deformed to $K[x]/p(x)$ where the degree of $p$ is $|C|$ where originally we had $x^{|C|}=1$. These are known as Ariki-Koike algebras and there is an extensive literature on these.

Even if you are only interested in hyperoctahedral groups you may find papers on Ariki-Koike algebras which give you what you want by specialising. For example, I have seen the semi-normal form for Ariki-Koike algebras but not for hyperoctahedral groups.

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As Bruce Westbury suggested at another question, the following book might be mentioned here.

A. V. Zelevinsky, Representations of Finite Classical Groups A Hopf algebra approach; Lecture Notes in Mathematics 869 (1981)

It is unfortunately probably hard to get anymore, and its typography is not very attractive, but I find the treatment very elegant. The representation theory of $S_n$ gives rise to an algebraic structure called Positive Self-Adjoint Hopf-algebra, whose simplest nontrivial model is the ring of symmetric functions with a comultiplication. Its combinatorial structure is entirely deduced from the axioms, and it provides models for among other things the representations rings for finite linear groups and for wreath products of the symmetric groups. Thus the hyperoctahedral groups are treated nicely as a simple special case.

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Would you be so kind to comment - what is comultiplication on symmetric functions ?  and where does it come from ? – Alexander Chervov Nov 26 2011 at 18:11
I like Zelevinsky's approach very much, but it diverges early from the usual treatments of the representation theory of the symmetric group, so I think it would be better to look at this after understanding one of the more straightforward treatments. – Tom Church Nov 26 2011 at 20:32
Symmetric functions are in infinitely many variables, and order doesn't matter. Now rename the variables $x_0,y_0,x_1,y_1,x_2,\ldots$ and decompose as a sum of products of a symmetric function in the $x$'s and one in the $y$'s. For instance $e_k$ gives $\sum_{i+j=k}e_i\otimes e_j$ since the monomials can be arbitrarily spread across the $x$'s and $y$'s, while $p_k$ gives $p_k\otimes1+1\otimes p_k$ since the monomials involve an $x_i$ or an $y_i$, but the two cannot mix in a power sum. – Marc van Leeuwen Nov 26 2011 at 20:33
@Tom: Yes I agree, this might not be the best introduction to the representations of the symmetric groups (especially if these are among the first groups you study representations of); a more concrete approach would be in place. But once you've seen a bit of that and you wonder if there is any higher perspective that explains why the details fall in place as they do, Zelevinsky's approach is a real revelation. – Marc van Leeuwen Nov 26 2011 at 20:41
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