I'm using the notation of Vilenkin and Klimyk, ''Part3: Representations of Lie Groups and Special Functions''', chapter 18.
Given an irreducible representation $T_m$ of the complex Lie algebra $U(n)$ with heighest weight $m = (m_{1n}, m_{2n}, \ldots, m_{nn})$ and $m_{1n} \geq m_{2n} \geq \ldots \geq m_{nn}$. The explicit actions of the representation of the Lie algebra of $U(n)$ is well known, see [VK, ch18.1.2, (5, 6, 7)].
I want to restrict the representation $T_m$ to the complex sub Lie algebra $U(n-1) \subset U(n)$. The representation $T_m$ restricted to $U(n-1)$ will decompose into irreducible representations $T_{m'}$ of $U(n-1)$ where $m_{j,n} \geq m'_{j,n-1} \geq m_{j+1, n}$ for all $1 \leq j \leq n-1$. In the same way the vector space $\mathfrak{H}_m$, generated by all Gelfand-Tsetlin patterns of $m = (m_{1n}, m_{2n}, \ldots, m_{nn})$, decomposes, uniquely up to a multiplication of basis elements, in $$ \mathfrak{H}_m = \bigoplus_{m_{j,n} \geq m'_{j,n-1} \geq m_{j+1, n}} \mathfrak{H}_{m'}, $$ where $\mathfrak{H}_{m'}$ is the vector space generated by all Gel'fand-Tsetlin patterns of $m' = (m'_{1,n-1}, m'_{2,n-1}, \ldots, m'_{n-1,n-1})$.
My question is if the (canonical) intertwiner between $\mathfrak{H}_m$ and $\bigoplus \mathfrak{H}_{m'}$ is explicitly known (probably in terms of special functions) for all $U(n-1) \subset U(n)$, $n \geq 2$?
If not, I will also be quite happy to know the explicit intertwiner for only $U(2) \subset U(3)$.
Additional keyword: Gel'fand-Tsetlin-patterns, intertwiners