Abstract
We compute all fusion algebras with symmetric rational S-matrix up to dimension 12. Only two of them may be used as S-matrices in a modular datum: the S-matrices of the quantum doubles of ℤ/2ℤ and S 3. Almost all of them satisfy a certain congruence which has some interesting implications, for example for their degrees. We also give explicitly an infinite sequence of modular data with rational S- and T-matrices which are neither tensor products of smaller modular data nor S-matrices of quantum doubles of finite groups. For some sequences of finite groups (certain subdirect products of S 3,D 4,Q 8,S 4), we prove the rationality of the S-matrices of their quantum doubles.
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Cuntz, M. Integral modular data and congruences. J Algebr Comb 29, 357–387 (2009). https://doi.org/10.1007/s10801-008-0139-y
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DOI: https://doi.org/10.1007/s10801-008-0139-y