\(\left.\text{HI(}\mathbb{Z}_3\right):\ \text{FR}^{6,2}_{8}\)
Fusion Rules
\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{5} & \mathbf{6} & \mathbf{4} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{4} & \mathbf{5} \\ \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{4} \ \mathbf{5} \ \mathbf{6}), (\mathbf{4} \ \mathbf{6} \ \mathbf{5}), (\mathbf{2} \ \mathbf{3}) (\mathbf{4} \ \mathbf{5})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(3.30278\) | \(\frac{1}{2} \left(3+\sqrt{13}\right)\) |
| \(\mathbf{5}\) | \(3.30278\) | \(\frac{1}{2} \left(3+\sqrt{13}\right)\) |
| \(\mathbf{6}\) | \(3.30278\) | \(\frac{1}{2} \left(3+\sqrt{13}\right)\) |
| \(\mathcal{D}_{FP}^2\) | \(35.725\) | \(3+\frac{3}{4} \left(3+\sqrt{13}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1 & 1 & 1 & \frac{1}{2} \left(3+\sqrt{13}\right) & \frac{1}{2} \left(3+\sqrt{13}\right) & \frac{1}{2} \left(3+\sqrt{13}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(3-\sqrt{13}\right) & \frac{1}{2} \left(3-\sqrt{13}\right) & \frac{1}{2} \left(3-\sqrt{13}\right) \\ 2 & -1 & -1 & 0 & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.000 & 3.303 & 3.303 & 3.303 \\ 1.000 & 1.000 & 1.000 & -0.3028 & -0.3028 & -0.3028 \\ 2.000 & -1.000 & -1.000 & 0 & 0 & 0 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
This fusion ring does not provide any representations of $SL_2(\mathbb{Z}).$
Adjoint Subring
The adjoint subring is the ring itself.
The upper central series is trivial.
Universal grading
This fusion ring allows only the trivial grading.
Categorifications
Data
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