\(D_4:\ \text{FR}^{8,2}_{1}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{3} & \mathbf{5} & \mathbf{4} \\ \mathbf{3} & \mathbf{6} & \mathbf{1} & \mathbf{7} & \mathbf{8} & \mathbf{2} & \mathbf{4} & \mathbf{5} \\ \mathbf{4} & \mathbf{7} & \mathbf{8} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{2} & \mathbf{3} \\ \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{2} \\ \mathbf{6} & \mathbf{3} & \mathbf{2} & \mathbf{5} & \mathbf{4} & \mathbf{1} & \mathbf{8} & \mathbf{7} \\ \mathbf{7} & \mathbf{4} & \mathbf{5} & \mathbf{3} & \mathbf{2} & \mathbf{8} & \mathbf{6} & \mathbf{1} \\ \mathbf{8} & \mathbf{5} & \mathbf{4} & \mathbf{2} & \mathbf{3} & \mathbf{7} & \mathbf{1} & \mathbf{6} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{4} \ \mathbf{3} \ \mathbf{5}), (\mathbf{2} \ \mathbf{5} \ \mathbf{3} \ \mathbf{4}), (\mathbf{2} \ \mathbf{3}) (\mathbf{7} \ \mathbf{8})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{3}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{5}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{6}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{6},\mathbf{7},\mathbf{8}\}\) | \(\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{6}\}\) | \(\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4},\mathbf{5},\mathbf{6}\}\) | \(\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(1.\) | \(1\) |
| \(\mathbf{6}\) | \(1.\) | \(1\) |
| \(\mathbf{7}\) | \(1.\) | \(1\) |
| \(\mathbf{8}\) | \(1.\) | \(1\) |
| \(\mathcal{D}_{FP}^2\) | \(8.\) | \(8\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{6} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 2 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{6} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.000 & 1.000 & -1.000 & -1.000 \\ 1.000 & 1.000 & -1.000 & -1.000 & -1.000 & -1.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.000 & -1.000 & -1.000 & -1.000 \\ 2.000 & -2.000 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
The adjoint subring is the trivial ring.
The upper central series is the following: \(D_4 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}', \text{deg}(\mathbf{4}) = \mathbf{4}', \text{deg}(\mathbf{5}) = \mathbf{5}', \text{deg}(\mathbf{6}) = \mathbf{6}', \text{deg}(\mathbf{7}) = \mathbf{7}', \text{deg}(\mathbf{8}) = \mathbf{8}'\), where the degrees form the group \(D_4\) with multiplication table:
\[\begin{array}{|llllllll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' & \mathbf{5}' & \mathbf{6}' & \mathbf{7}' & \mathbf{8}' \\ \mathbf{2}' & \mathbf{1}' & \mathbf{6}' & \mathbf{8}' & \mathbf{7}' & \mathbf{3}' & \mathbf{5}' & \mathbf{4}' \\ \mathbf{3}' & \mathbf{6}' & \mathbf{1}' & \mathbf{7}' & \mathbf{8}' & \mathbf{2}' & \mathbf{4}' & \mathbf{5}' \\ \mathbf{4}' & \mathbf{7}' & \mathbf{8}' & \mathbf{1}' & \mathbf{6}' & \mathbf{5}' & \mathbf{2}' & \mathbf{3}' \\ \mathbf{5}' & \mathbf{8}' & \mathbf{7}' & \mathbf{6}' & \mathbf{1}' & \mathbf{4}' & \mathbf{3}' & \mathbf{2}' \\ \mathbf{6}' & \mathbf{3}' & \mathbf{2}' & \mathbf{5}' & \mathbf{4}' & \mathbf{1}' & \mathbf{8}' & \mathbf{7}' \\ \mathbf{7}' & \mathbf{4}' & \mathbf{5}' & \mathbf{3}' & \mathbf{2}' & \mathbf{8}' & \mathbf{6}' & \mathbf{1}' \\ \mathbf{8}' & \mathbf{5}' & \mathbf{4}' & \mathbf{2}' & \mathbf{3}' & \mathbf{7}' & \mathbf{1}' & \mathbf{6}' \\ \hline \end{array}\]Categorifications
Data
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