Let $G=\mathrm U_n$ or $\mathrm{GL}_n(\mathbf C)$ and $H$ the subgroup of block diagonal matrices respecting a partition $n=n_1+\dots+n_k$. Is the normalizer $N=N_G(H)$ computed anywhere in the literature?

I guess, but haven’t proved, that it is generated by $H$ and the permutations (“transpositions”) exchanging the partition’s same-length segments ($\smash{n_i=n_j}$, if any). I also suspect this may be discussed in these papers, to which I don’t have access:

*Koĭbaev, V. A.*, On subgroups of the full linear groups containing a group of elementary block diagonal matrices. ZBL0521.20027. (1982, translation 1983; other translation?)*Borevich, Z. I.; Vavilov, N. A.*, Ordering of subgroups, containing a group of block-diagonal matrices, in the general linear group over a ring. ZBL0512.20031. (1982; translation?)*Vavilov, N. A.*, Subgroups of the general linear group over a semi-local ring containing the group of block-diagonal matrices. ZBL0509.20035. (1983; ever translated?)