How does the spectra of $DU$ change when $D$ runs over all diagonal unitary matrices? Here $U$ is a fixed unitary matrix. Precisely, let spec$(X)$ be a set of eigenvalues of $X$. For a unitary matrix $U$ let $$ S_U=\{ \mbox{spec}(DU)\;|\;D\mbox{ runs over all diagonal unitary matrices}\}. $$ Clearly, $S_U$ depends on $U$. Two examples:

if $U$ is diagonal then spec($DU$) run over all possibilities: $S_U=({\mathbb C}_1)^n$, where ${\mathbb C}_1$ is the set of complex numbers of norm $1$ and $n$ is the size of $U$.

Let $U$ be a cyclic permutation, that is

$$ U=\left(\begin{array}{cccc} 0 & 0 & 0\dots\dots & 1\\\ 1 & 0 & 0\dots\dots & 0\\\ 0 & 1 & 0\dots\dots & 0\\\ \dots & \dots & \dots\dots\dots &\dots \\\ \end{array}\right). $$ Then $S_U=\{\{wa,w^2a,w^3a,...,w^na\}\;\;|\;\;a\in{\mathbb C}_1,\;w \mbox{ is primitive root of 1}\}$

Is something known about $S_U$ in general?

What is $S_U$ for

a) random $U$

b)for which $U$ $S_U$ looks like in example 2.

c) some concrete $U$, say $U$ being a finite Fourier transform matrices?