  Building on work of M. M\"uger 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.
