On the Casimir number and formal codegree of Haagerup-Izumi fusion rings

For any cyclic group Zₙ, we first determine the Casimir number and determinant of the Haagerup-Izumi fusion ring HIZₙ, it turns out that they do not share the same set of prime factors. Then we show that all finite-dimensional irreducible representations of HIZₙ are defined over certain cyclotomic fields. As a direct result, we obtain the formal codegrees of HIZₙ, which satisfy the pseudo-unitary inequality.