\(\text{FR}^{8,4}_{17}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{7} & \mathbf{8} & \mathbf{5} & \mathbf{6} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} \\ \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{5} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6}), (\mathbf{2} \ \mathbf{3}) (\mathbf{7} \ \mathbf{8}), (\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{7} \ \mathbf{6} \ \mathbf{8}), (\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{8} \ \mathbf{6} \ \mathbf{7})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{3}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathbf{6}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathbf{7}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathbf{8}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathcal{D}_{FP}^2\) | \(75.7771\) | \(4+4 \left(2+\sqrt{5}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{8} & \mathbf{5} & \mathbf{7} \\ \hline 1 & 1 & 1 & 1 & 2+\sqrt{5} & 2+\sqrt{5} & 2+\sqrt{5} & 2+\sqrt{5} \\ 1 & 1 & 1 & 1 & 2-\sqrt{5} & 2-\sqrt{5} & 2-\sqrt{5} & 2-\sqrt{5} \\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\ 1 & 1 & -1 & -1 & i & -i & -i & i \\ 1 & 1 & -1 & -1 & -i & i & i & -i \\ 1 & -1 & -1 & 1 & i & i & -i & -i \\ 1 & -1 & -1 & 1 & -i & -i & i & i \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{8} & \mathbf{5} & \mathbf{7} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 4.236 & 4.236 & 4.236 & 4.236 \\ 1.000 & 1.000 & 1.000 & 1.000 & -0.2361 & -0.2361 & -0.2361 & -0.2361 \\ 1.000 & -1.000 & 1.000 & -1.000 & 1.000 & -1.000 & 1.000 & -1.000 \\ 1.000 & -1.000 & 1.000 & -1.000 & -1.000 & 1.000 & -1.000 & 1.000 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.000 i & -1.000 i & -1.000 i & 1.000 i \\ 1.000 & 1.000 & -1.000 & -1.000 & -1.000 i & 1.000 i & 1.000 i & -1.000 i \\ 1.000 & -1.000 & -1.000 & 1.000 & 1.000 i & 1.000 i & -1.000 i & -1.000 i \\ 1.000 & -1.000 & -1.000 & 1.000 & -1.000 i & -1.000 i & 1.000 i & 1.000 i \\ \hline \end{array}\]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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