Fourier matrices of small rank

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.

Keywords

• Fourier matrices,Modular data,Fusion rings,C-algebras

References

1. [1] Z. Arad, E. Fisman, M. Muzychuk, Generalized table algebras, Israel J. Math. 114(1) (1999) 29–60.
2. [2] H. I. Blau, Table algebras, European J. Combin. 30(6) (2009) 1426–1455.
3. [3] M. Cuntz, Integral modular data and congruences, J. Algebraic Combin. 29(3) (2009) 357–387.
4. [4] P. Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Springer–Verlag, New York, 1997.
5. [5] T. Gannon, Modular data: The algebraic combinatorics of conformal field theory, J. Algebraic Combin. 22(2) (2005) 211–250.
6. [6] A. Hanaki, I. Miyamoto, Classification of association schemes with small vertices, 2017, available at: math.shinshu-u.ac.jp/ hanaki/as/.
7. [7] D. G. Higman, Coherent algebras, Linear Algebra Appl. 93 (1987) 209–239.
8. [8] J. D. Qualls, Lectures on Conformal Field Theory, arXiv:1511.04074 [hep-th].

9. [9] E. L. Rees, Graphical Discussion of the Roots of a Quartic Equation, Amer. Math. Monthly 29(2) (1922) 51–55.
10. [10] M. Schottenloher, A Mathematical Introduction to Conformal Field Theory, Springer–Verlag, Berlin, Heidelberg, 2nd edition, 2008.
11. [11] G. Singh, Classification of homogeneous Fourier matrices, arXiv:1610.05353 [math.RA].
12. [12] B. Xu, Characters of table algebras and applications to association schemes, J. Combin. Theory Ser. A 115(8) (2008) 1358–1373.
13. [13] A. Zahabi, Applications of Conformal Field Theory and String Theory in Statistical Systems, Ph.D. dissertation, University of Helsinki, Helsinki, Finland, 2013.