The hyperoctahedral group $H_n$ has several descriptions; as a wreath product; as signed permutation matrices; as the Weyl group of type $B_n$ or $C_n$. In all these descriptions it is apparent that the symmetric group $S_n$ is a subgroup.

I would like to know the Frobenius characters of the restrictions of the irreducible characters of $H_n$ to $S_n$. I imagine this is known but not to me.

The irreducible characters of $H_n$ are indexed by pairs of partitions $\alpha$,$\beta$ such that the total number of boxes is $n$. There is a well developed combinatorial theory involving bitableaux as well as an analogue of the Robinson-Schensted correspondence. The character theory is described in I. G. MacDonald "Symmetric functions and Hall polynomials" Chapter I, Appendix B in terms of symmetric functions.

I am also aware of the question 48532 which asks about the restriction map for the inclusion of $H_n$ in $S_{2n}$.

  • $\begingroup$ Having posted the question I now see this is simply $s_\alpha s_\beta$. This can be seen from MacDonald as if we set the two sets of variables equal the change of variable does nothing. $\endgroup$ Nov 24 '11 at 21:02

Even though you found the answer, I'd like to point to the following reference, where I learned this result, at a time when the book by Macdonald (note the captitalization) did not yet have an appendix B to chapter I (i.e., before its second edition):

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

The answer to your question is contained in the Proposition in section 7.10 (page 105), which says (somewhat indirectly) that more generally for the wreath product of $S_n$ and a finite abelian group $G$, the irreducible representations correspond to families of partitions indexed by the irreducible representations of $G$ and of total size $n$, and that the restriction to $S_n$ of such a representation is obtained by multiplying together, in the combined Grothendieck ring $R(S)$ of the symmetric groups, the irreducible representations associated to those partitions. The multiplication in $R(S)$ is defined by induction from a product of small symmetric groups into the containing symmetric group $S_n$, and so a the level of Frobenius characters becomes multiplication of Schur functions.

  • $\begingroup$ Thanks. That is also very close to a nice answer to question 25625. $\endgroup$ Nov 25 '11 at 13:48

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