Classification of homogeneous Fourier matrices

Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group $\mathrm{SL}_2(\mathbb{Z})$. In this paper, we show that there is a one-to-one correspondence between Fourier matrices associated to modular data and self-dual $C$-algebras that satisfy a certain condition. We prove that a homogenous $C$-algebra arising from a Fourier matrix has all the degrees equal to $1$.