Fourier matrices of small rank
Modular data is an important topic of study in rational conformal field theory. Cuntz, using a computer, classified the Fourier matrices associated to modular data with rational entries up to rank $12$, see . Here we use the properties of $C$-algebras arising from Fourier matrices to classify complex Fourier matrices under certain conditions up to rank $5$. Also, we establish some results that are helpful in recognizing $C$-algebras that not arising from Fourier matrices by just looking at the first row of their character tables.
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