# Restriction of characters of hyperoctahedral groups.

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 $S_{2n}$.

• 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. Nov 24 '11 at 21:02

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.