Asymptotic density of states in 2d CFTs with non-invertible symmetries

It is known that the asymptotic density of states of a 2d CFT in an irreducible representation ρ of a finite symmetry group G is proportional to (dim ρ)². We show how this statement can be generalized when the symmetry can be non-invertible and is described by a fusion category $$ \mathcal{C} $$C . Along the way, we explain what plays the role of a representation of a group in the case of a fusion category symmetry; the answer to this question is already available in the broader mathematical physics literature but not yet widely known in hep-th. This understanding immediately implies a selection rule on the correlation functions, and also allows us to derive the asymptotic density.