Modular data (MD) is an invariant of a modular fusion category $\mathcal{C}$. It is a couple of symmetric matrices $S, T \in M_r(\mathbb{C})$ with:
- $r$ the rank of $\mathcal{C}$,
- $S$ invertible,
- $T$ unitary, diagonal and of finite order,
- $(ST)^3 = S^2$ and $S^4 = 1$, up to multiplicative constants.
By the last point, a MD gives a projective representation of the modular group $SL_2(\mathbb{Z})$. For more details, see Theorem 2.1 in [NRWW], where Table 1 lists all $45$ MD up to rank $5$. We can observe that for such low rank, the MD is completely determined by the T-matrix. I expect that it is not true in general, but I would like to know counter-examples.
Question: What are examples (if any) of two different MD with same T-matrix?
Here different should be understood up to a permutation of the basis.
[NRWW] Siu-Hung Ng, Eric C Rowell, Zhenghan Wang, Xiao-Gang Wen, Reconstruction of modular data from SL2(Z) representations, arXiv:2203.14829v1.