Inner functions of matrix argument and conjugacy classes in unitary groups

Denote by $B_n$ the set of complex square matrices order $n$, whose Euclidean operator norms are $<1$. Its Shilov boundary is $U(n)$ all unitary matrices. A holomorphic map $B_m\to B_n$ inner if it sends $U(m)$ to $U(n)$. On other hand we consider a group $U(n+mj)$ and its subgroup $U(j)$ embedded in block-diagonal way ($m$ blocks unit block size $n$). For any conjugacy class with respect assign 'characteristic function', which rational B_n$. We show that functions, can be obtained as functions', closed natural operations pointwise direct sums, products, compositions, substitutions finite-dimensional representations general linear groups, etc. also describe explicitly corresponding on classes.