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

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