\(\text{FR}^{8,4}_{11}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{5} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{6} \\ \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{1}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{2}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{5}+\mathbf{6} \\ \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{5} & \mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6}), (\mathbf{3} \ \mathbf{4}) (\mathbf{7} \ \mathbf{8})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{7},\mathbf{8}\}\) | \(\text{Pseudo PSU(2})_6:\ \text{FR}^{4,2}_{4}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(2.41421\) | \(1+\sqrt{2}\) |
| \(\mathbf{6}\) | \(2.41421\) | \(1+\sqrt{2}\) |
| \(\mathbf{7}\) | \(2.41421\) | \(1+\sqrt{2}\) |
| \(\mathbf{8}\) | \(2.41421\) | \(1+\sqrt{2}\) |
| \(\mathcal{D}_{FP}^2\) | \(27.3137\) | \(4+4 \left(1+\sqrt{2}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{6} \\ \hline 1 & 1 & 1 & 1 & 1+\sqrt{2} & 1+\sqrt{2} & 1+\sqrt{2} & 1+\sqrt{2} \\ 1 & 1 & 1 & 1 & 1-\sqrt{2} & 1-\sqrt{2} & 1-\sqrt{2} & 1-\sqrt{2} \\ 1 & 1 & -1 & -1 & -1-\sqrt{2} & 1+\sqrt{2} & 1+\sqrt{2} & -1-\sqrt{2} \\ 1 & 1 & -1 & -1 & \sqrt{2}-1 & 1-\sqrt{2} & 1-\sqrt{2} & \sqrt{2}-1 \\ 1 & -1 & -i & i & 1 & i & -i & -1 \\ 1 & -1 & i & -i & 1 & -i & i & -1 \\ 1 & -1 & i & -i & -1 & i & -i & 1 \\ 1 & -1 & -i & i & -1 & -i & i & 1 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.414 & 2.414 & 2.414 & 2.414 \\ 1.000 & 1.000 & 1.000 & 1.000 & -0.4142 & -0.4142 & -0.4142 & -0.4142 \\ 1.000 & 1.000 & -1.000 & -1.000 & -2.414 & 2.414 & 2.414 & -2.414 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0.4142 & -0.4142 & -0.4142 & 0.4142 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 1.000 & 1.000 i & -1.000 i & -1.000 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 1.000 & -1.000 i & 1.000 i & -1.000 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & -1.000 & 1.000 i & -1.000 i & 1.000 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & -1.000 & -1.000 i & 1.000 i & 1.000 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
This fusion ring does not provide any representations of $SL_2(\mathbb{Z}).$
Adjoint Subring
Elements \(\mathbf{1}, \mathbf{2}, \mathbf{7}, \mathbf{8}\), form the adjoint subring \(\text{Pseudo PSU(2})_6:\ \text{FR}^{4,2}_{4}\) .
The upper central series is the following: \(\text{FR}^{8,4}_{11} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{7}, \mathbf{8} }{\supset} \text{Pseudo PSU(2})_6\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{2}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{1}', \text{deg}(\mathbf{8}) = \mathbf{1}'\), where the degrees form the group \(\mathbb{Z}_2\) with multiplication table:
\[\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}\]Categorifications
Data
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