Integral modular data and congruences

Details Integral modular data and congruences Integral modular data and congruences

2006-11-08

arxiv.org/abs/math/0611233v2

Mathematics

Representation Theory

27 pages

Scientific paper

10.1007/s10801-008-0139-y

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 $\mathbb{Z}/2\mathbb{Z}$ 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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