Fourier matrices of small rank
Keywords:
Fourier matrices, Modular data, Fusion rings, C-algebras
Abstract
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 [3]. 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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D. G. Higman, Coherent algebras, Linear Algebra Appl. 93 (1987) 209–239.
J. D. Qualls, Lectures on Conformal Field Theory, arXiv:1511.04074 [hep-th].
E. L. Rees, Graphical Discussion of the Roots of a Quartic Equation, Amer. Math. Monthly 29(2) (1922) 51–55.
M. Schottenloher, A Mathematical Introduction to Conformal Field Theory, Springer–Verlag, Berlin, Heidelberg, 2nd edition, 2008.
G. Singh, Classification of homogeneous Fourier matrices, arXiv:1610.05353 [math.RA].
B. Xu, Characters of table algebras and applications to association schemes, J. Combin. Theory Ser. A 115(8) (2008) 1358–1373.
A. Zahabi, Applications of Conformal Field Theory and String Theory in Statistical Systems, Ph.D. dissertation, University of Helsinki, Helsinki, Finland, 2013.
