Classification of non-homogeneous Fourier matrices associated with modular data up to rank 5

Modular data is an important topic of study in rational conformal field theory. A modular datum defines finite dimensional representations of the modular group $\mbox{SL}_2(\mathbf{Z})$. For every Fourier matrix in a modular datum there exists an Allen matrix obtained from the Fourier matrix after dividing each its row with the first entry of that row. In this paper, we classify the non-homogenous Fourier matrices and non-homogenous Allen matrices up to rank $5$. The methods developed here are useful for the classification of the matrices of higher ranks. Also, we establish some results that are helpful in recognizing the $C$-algebras not arising from Allen matrices by just looking at the character table of the $C$-algebra, in particular, the first row of the character table.