On the Mathieu Conjecture for $SU(N)$ and $SO(N)$

Building on work of M. Müger and L. Tuset, we reduce the Mathieu conjecture, formulated by O. Mathieu in 1997, for $SU(N)$ to a simpler conjecture in purely abelian terms. We sketch a similar reduction for $SO(N)$. The proofs rely on Euler-style parametrizations of these groups, which we discuss including proofs.