Embed the hyperoctahedral group $H\_n$ into the symmetric group $S\_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S\_{2n}$ by partitions of $2n$ and label representations of $H\_n$ by pairs of partitions whose sizes add up to $n$ in the standard way. I am looking for a combinatorial description of the branching rule from $S\_{2n}$ to $H\_n$. This should be in the literature, but I couldn't find it.


Second small question: Is it possible to have a different embedding of $H\_n$ into $S\_{2n}$ so that the rule changes? I'd guess probably, but didn't think hard about it.



Branching rule from symmetric group $S_{2n}$ to hyperoctahedral group $H_n$
Okay I found it in the article by Koike and Terada, "Littlewood's formulas and their application to representations of classical Weyl groups". Here is the theorem. Define $d^\nu\_{\lambda, \mu}$ as the coefficients in the product of plethysms:


$\displaystyle (s\_\lambda \circ h\_2)(s\_\mu \circ e\_2) = \sum\_\nu d^\nu\_{\lambda, \mu} s\_\nu$


where $s\_\lambda$ is a Schur function, $h\_2$ is the complete homogeneous symmetric function of degree 2, and $e\_2$ is the elementary symmetric function of degree 2. Then the restriction of the representation $\chi(\nu)$ of $S\_{2n}$ to $H\_n$ is $\sum\_{\lambda, \mu} d^\nu\_{\lambda, \mu} \chi(\lambda, \mu)$.


Richard Stanley has informed me that a more general statement appears as Theorem A2.8 of *Enumerative Combinatorics Vol. 2* (set $k=2$). This formula gives the coefficients for representations of $H\_n$ indexed by bipartitions where one of the partitions is the empty set. The general case comes from the fact that $\chi(\lambda, \mu)$ is induced from a parabolic subgroup $H\_k \times H\_{n-k}$ of $H\_n$ and that the map ch takes induction to product of characters.


