We know that Schur functions $S_\lambda$ are related with irreps. of $U(n)$ and that there is an associated branching rule for the subgroup chain $U(1)\subseteq \ldots \subseteq U(n-1)\subseteq U(n)$, namely
$$S_\lambda(x_1,...,x_n)=\sum_\rho S_\rho(x_1,...,x_{n-1})x_n^{|\lambda|-|\rho|},$$ with $|\rho|$ the size of the partition $\rho$. There is a way of inverting this process? that is, having $S_\lambda(x_1,...,x_k)$ and $S_\rho(x_{k+1},...,x_{n})$ it is possible to construct a representation for $U(n)$ like $$S_\lambda(x_1,...,x_k)S_\rho(x_{k+1},...,x_{n})=\sum_\gamma a_\gamma S_\gamma (x_1,...,x_n),$$ or any other way of lifting a $U(m)$ representation onto a $U(n)$, for $m<n$?