7
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.