On Euler angles decomposition of $\mathrm{SU}(N)$

Question

$\DeclareMathOperator\SU{SU}$I am looking for a (generalized) Euler angles decomposition for $\SU(N)\ (N>1)$ in the following fashion: $$ \SU(N)\ni m = a\, u \, b $$ where $a,b$ are independent diagonal $\SU(N)$-matrices each of which accounts for $N-1$ parameters while $u\in \SU(N)$ is parametrized by the remain $(N-1)^2$ parameters. For instance the matrices $u$ might form a $U(N)$-isomorphic subgroup of $\SU(N)$. Notice that in the case $N=2$ this decomposition reduces to the known Euler's one $a(b)= \exp(i \alpha(\beta) \sigma_3)$ and $U = \exp(i \gamma \sigma_2)$ where $\sigma_j$ is a Pauli matrix and $\alpha$, $\beta$, $\gamma$ are the angles.

I found in literature Bertini, Cacciatori, and Cerchiai - On the Euler angles for $\operatorname{SU}(N)$ and Tilma and Sudarshan - Generalized Euler angle parametrization for $\operatorname{SU}(N)$ other kinds of generalizations of Euler angles decompositions, however they differ in form from the one above.

Can you point me to some relevant literature? Can you show me a working parametrization of $u$ such that the whole $\SU(N)$ is covered?

Answer 1

Building on the paper Idel and Wolf - Sinkhorn normal form for unitary matrices Colin McQuillan suggested, it is easy to see that every $\operatorname{SU}(N)$ matrix $m$ can be decomposed as

$$ m = a \,u\, b $$ with $a,b \in \operatorname{SU}(N)$ and diagonal, and
$$ u = F \,\left(\frac{1}{\mathrm{det}(V)} \oplus V\right) F^\dagger \in \operatorname{SU}(N) $$

where $F_{jk} = \frac{1}{\sqrt{N}}e^{\frac{2\pi i}{N}jk}$ is the Fourier transform unitary matrix and $V\in \operatorname{U}(N-1)$.

Notice that the number of parameters of this factorization matches the dimension of $\operatorname{SU}(N)$.