Littlewood-Richardson-Type Rule for Restriction from $S_{2n}$ to $S_{2(n-t)} \times (S_2 \wr S_t)$

Question

It is well-known that the Littlewood-Richardson coefficient $c^{\nu}_{\lambda \mu}$ is the number of times the irreducible representation $V_\lambda \bigotimes V_\mu$ of the product of symmetric groups $S_{|\lambda|} × S_{|\mu|}$ appears in the restriction of the representation $V_{\nu}$ of $S_{|\nu|}$ to $S_{|\lambda|} × S_{|\mu|}$.

It is also known that the irreducible representations of the hyperoctahedral group $H_t := (S_2 \wr S_t) \leq S_{2t}$ are indexed by pairs of partitions $(\alpha,\beta)$ such that $|\alpha| + |\beta| = t$.

Let $\nu \vdash 2n$, $\lambda \vdash 2(n-t)$, and let $c^{\nu}_{\lambda(\alpha,\beta)}$ be the number of times the irreducible representation $V_\lambda \bigotimes V_{(\alpha,\beta)}$ of the $S_{|\lambda|} × H_t$ appears in the restriction of the representation $V_{\nu}$ of $S_{2n}$ to $S_{|\lambda|} \times H_t$. My question is the following:

Is there a branching/Littlewood-Richardson-type rule for computing $c^{\nu}_{\lambda(\alpha,\beta)}$ in the literature?

By the Littlewood-Richardson rule, transitivity of induction, and Frobenius reciprocity, a branching rule for restricting from $S_{2(n-t)} \times S_{2t}$ to $S_{2(n-t)} \times H_t$ would also do the trick. This is quite similar to an older post of Steven Sam, but unfortunately I do not have access to the reference that answers his question.

I might add that I am really only interested in whether the trivial representation of $S_{|\lambda|} \times H_t$ appears when an irreducible representation $V_{\nu}$ of $S_{2n}$ is restricted to $S_{|\lambda|} \times H_t$. I'm not a rep. theorist/algebraist, so if there is some easier way of determining this, I would be happy to know.

Answer 1

For the trivial representation the problem is much simplified. Let $1_G$ denote the trivial character of a group $G$. It is well known that $1_{S_2 \wr S_t}\!\!\uparrow^{S_{2t}} = \sum_{\mu \vdash t} \chi^{2\mu}$ where $2\mu$ is the partition obtained from $\mu$ by doubling the length of each part.

By two applications of Frobenius reciprocity, we have

$$ \begin{align*} \bigl\langle \chi^\nu \!\!\downarrow_{S_{2(n-t)} \times S_2 \wr S_t}, 1_{S_{2(n-t)}} \times 1_{S_2 \wr S_t} \bigr\rangle &= \bigl\langle \chi^\nu \!\!\downarrow_{S_{2(n-t)} \times S_{2t}}, 1_{S_{2(n-t)}} \times 1_{S_2 \wr S_t}\!\!\uparrow^{S_{2t}} \bigr\rangle \\ &= \bigl\langle \chi^\nu, \sum_{\mu \vdash t} \bigl( 1_{S_{2(n-t)}} \times \chi^{2\mu} \bigr)\!\!\uparrow^{S_{2n}} \bigr\rangle \end{align*} $$

Hence by Young's rule (or the more general Littlewood—Richardson rule), the multiplicity of the trivial character in $\chi^\nu \downarrow_{S_{2(n-t)} \times S_2 \wr S_t}$ is equal to the number of partitions $\mu$ of $t$ such that the Young diagram of $\nu$ can be obtained from the Young diagram of $2\mu$ by adding $2(n-t)$ boxes, no two in the same column.