  We discuss a class of selection rules which i) do not come from group actions
on fields, ii) are exact at tree level in perturbation theory, iii) are
increasingly violated as the loop order is raised, and iv) eventually reduce to
selection rules associated with an ordinary group symmetry. We start from basic
field-theoretical examples in which fields are labeled by conjugacy classes
rather than representations of a group, and discuss generalizations using
fusion algebras or hypergroups. We also discuss how such selection rules arise
naturally in string theory, such as for non-Abelian orbifolds or other cases
with non-invertible worldsheet symmetries.
