$U(n)$ is the group of $n$-dimensional complex unitary matrices. This group has irreducible representations in higher dimensions. Let $R_{n,d}$ be an irrep of $U(n)$ in terms of $d$-dimensional matrices, $d>n$. The matrices in $R_{n,d}$ are unitary, so this is a subgroup of $U(d)$, $$R_{n,d}\subset U(d).$$
What is known about the way $R_{n,d}$ sits inside $U(d)$? For example, what is $U(d)/R_{n,d}$?
