I'm inexperienced in the representation theory of the symmetric group, so please correct my possible mistakes. Fix $m\leq n$, $G:=S_n$ and $H:=S_m\times S_{n-m}$ as a Young subgroup of $G$. Let $V^{\lambda}_i$ denote the irreducible $S_i$-module associated to partition $\lambda\vdash i$. By the Littlewood-Richardson rule, we have $$\text{Ind}_H^G(V_m^{\lambda}\otimes V_{n-m}^{\mu}) \cong \bigoplus_{\nu\vdash n} c^\nu_{\lambda,\mu} V_n^\nu $$ where $c^\nu_{\lambda,\mu}$ are the Littlewood-Richardson coefficients.
By Frobenius reciprocity, we also have that $$\text{Res}_G^H( V_n^\nu) \cong \bigoplus_{\sigma\vdash m,\tau\vdash n-m} c^\nu_{\sigma,\tau} V_m^{\sigma}\otimes V_{n-m}^{\tau}.$$
Therefore my representation of interest, which is the restriction to $H$ of the induced $G$-representation of an irreducible $H$-module, has the irreducible decomposition$$\text{Res}_G^H\text{Ind}_H^G(V_m^{\lambda}\otimes V_{n-m}^{\mu}) \cong \bigoplus_{\sigma\vdash m,\tau\vdash n-m}(\sum_{\nu\vdash n} c^\nu_{\lambda,\mu}c^\nu_{\sigma,\tau})V_m^{\sigma}\otimes V_{n-m}^{\tau}.$$
- Is the above decomposition right?
- If it is, do the numbers $b_{\lambda,\mu,\sigma,\tau}:=\sum_{\nu\vdash n} c^\nu_{\lambda,\mu}c^\nu_{\sigma,\tau}$ have any name? Do they have any combinatorial characterization, or can we compute them in any other way?
