  We go beyond a systematic review of the semiclassical approaches for
determining the scaling dimensions of fixed-charge operators in $U(1)$ and
$O(N)$ models by introducing a general strategy apt at determining the relation
between a given charge configuration and the associated operators for more
involved symmetry groups such as the $U(N) \times U(M)$. We show how, varying
the charge configuration, it is possible to access anomalous dimensions of
different operators transforming according to a variety of irreducible
representations of the non-abelian symmetry group without the aid of
diagrammatical computations. We illustrate our computational strategy by
determining the anomalous dimensions of several composite operators to the
next-to-leading order in the semiclassical expansion for the $U(N) \times U(M)$
conformal field theory (CFT) in $4-\epsilon$ dimensions. Thanks to the powerful
interplay between semiclassical methods and group theory we can, for the first
time, extract scaling dimensions for a wide range of operators.
