This paper studies a new source of selection rules in quantum field theories that comes from gauging a discrete group of automorphisms. Start with a theory that has a discrete global symmetry G. Then take another discrete group H that acts on G by automorphisms (this is like the symmetry operations of an orbifold). When the authors “gauge” H — that is, restrict to H‑invariant states — the remaining symmetry constraints on allowed interactions are not always described by an ordinary group. Instead they produce non‑invertible selection rules: rules that forbid or allow interactions but do not come from a usual symmetry group with inverse elements for every operation.

To describe these models the authors build a general framework that includes generalized field transformations under H. In practice a field is labeled by how it transforms under G (by an irreducible representation, the basic building block of symmetry action). The H action can mix different components inside those multiplets when H is non‑Abelian. That mixing can remove some components of the multiplet that would otherwise produce a trivial singlet in a tensor product. As a result, the usual checks for allowed couplings — based on tensor product decompositions or conjugacy classes of G — can be necessary but not sufficient.

To get a complete criterion the paper analyzes the full semidirect product G ⋊ H and introduces a quantity called a projected character. A projected character is roughly the trace of a representation but restricted to the H‑invariant part of the representation space. Using these projected characters the authors give a necessary and sufficient condition for an n‑point bare coupling to be nonzero. They also show that the surviving field components obey an associative, fusion‑like algebra controlled by Clebsch‑Gordan coefficients — the numbers that tell you how products of multiplets split into pieces.