\(\mathbb{Z}_2 \times TY(\mathbb{Z}_3):\ \text{FR}^{8,4}_{3}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{6} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \mathbf{4} & \mathbf{5} & \mathbf{1} & \mathbf{3} & \mathbf{6} & \mathbf{2} & \mathbf{7} & \mathbf{8} \\ \mathbf{5} & \mathbf{4} & \mathbf{2} & \mathbf{6} & \mathbf{3} & \mathbf{1} & \mathbf{8} & \mathbf{7} \\ \mathbf{6} & \mathbf{3} & \mathbf{5} & \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{8} & \mathbf{7} \\ \mathbf{7} & \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4} & \mathbf{2}+\mathbf{5}+\mathbf{6} \\ \mathbf{8} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{2}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{3}+\mathbf{4} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{7} \ \mathbf{8}), (\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
| \(\{\mathbf{1},\mathbf{3},\mathbf{4},\mathbf{7}\}\) | \(\text{Potts}:\ \text{FR}^{4,2}_{2}\) |
| \(\{\mathbf{1},\mathbf{3},\mathbf{4},\mathbf{8}\}\) | \(\text{Potts}:\ \text{FR}^{4,2}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\}\) | \(\mathbb{Z}_6:\ \text{FR}^{6,4}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(1.\) | \(1\) |
| \(\mathbf{6}\) | \(1.\) | \(1\) |
| \(\mathbf{7}\) | \(1.73205\) | \(\sqrt{3}\) |
| \(\mathbf{8}\) | \(1.73205\) | \(\sqrt{3}\) |
| \(\mathcal{D}_{FP}^2\) | \(12.\) | \(12\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1 & -1 & 1 & 1 & -1 & -1 & -\sqrt{3} & \sqrt{3} \\ 1 & -1 & 1 & 1 & -1 & -1 & \sqrt{3} & -\sqrt{3} \\ 1 & 1 & 1 & 1 & 1 & 1 & -\sqrt{3} & -\sqrt{3} \\ 1 & 1 & 1 & 1 & 1 & 1 & \sqrt{3} & \sqrt{3} \\ 1 & 1 & e^{\frac{2 i \pi }{3}} & e^{-\frac{2 i \pi }{3}} & e^{-\frac{2 i \pi }{3}} & e^{\frac{2 i \pi }{3}} & 0 & 0 \\ 1 & -1 & e^{\frac{2 i \pi }{3}} & e^{-\frac{2 i \pi }{3}} & e^{\frac{i \pi }{3}} & e^{-\frac{i \pi }{3}} & 0 & 0 \\ 1 & -1 & e^{-\frac{2 i \pi }{3}} & e^{\frac{2 i \pi }{3}} & e^{-\frac{i \pi }{3}} & e^{\frac{i \pi }{3}} & 0 & 0 \\ 1 & 1 & e^{-\frac{2 i \pi }{3}} & e^{\frac{2 i \pi }{3}} & e^{\frac{2 i \pi }{3}} & e^{-\frac{2 i \pi }{3}} & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1. & -1. & 1. & 1. & -1. & -1. & -1.73205 & 1.73205 \\ 1. & -1. & 1. & 1. & -1. & -1. & 1.73205 & -1.73205 \\ 1. & 1. & 1. & 1. & 1. & 1. & -1.73205 & -1.73205 \\ 1. & 1. & 1. & 1. & 1. & 1. & 1.73205 & 1.73205 \\ 1. & 1. & -0.5+0.866025 i & -0.5-0.866025 i & -0.5-0.866025 i & -0.5+0.866025 i & 0. & 0. \\ 1. & -1. & -0.5+0.866025 i & -0.5-0.866025 i & 0.5 +0.866025 i & 0.5 -0.866025 i & 0. & 0. \\ 1. & -1. & -0.5-0.866025 i & -0.5+0.866025 i & 0.5 -0.866025 i & 0.5 +0.866025 i & 0. & 0. \\ 1. & 1. & -0.5-0.866025 i & -0.5+0.866025 i & -0.5+0.866025 i & -0.5-0.866025 i & 0. & 0. \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
This fusion ring does not provide any representations of $SL_2(\mathbb{Z}).$
Adjoint Subring
Elements \(\mathbf{1}, \mathbf{3}, \mathbf{4}\), form the adjoint subring \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) .
The upper central series is the following: \(\mathbb{Z}_2\text{$\times $Potts} \underset{ \mathbf{1}, \mathbf{3}, \mathbf{4} }{\supset} \mathbb{Z}_3 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{2}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{3}', \text{deg}(\mathbf{8}) = \mathbf{4}'\), where the degrees form the group \(\mathbb{Z}_2\times \mathbb{Z}_2\) with multiplication table:
\[\begin{array}{|llll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' \\ \mathbf{2}' & \mathbf{1}' & \mathbf{4}' & \mathbf{3}' \\ \mathbf{3}' & \mathbf{4}' & \mathbf{1}' & \mathbf{2}' \\ \mathbf{4}' & \mathbf{3}' & \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}\]Categorifications
Data
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