\(\text{FR}^{7,2}_{12}\)
Fusion Rules
\[\begin{array}{|lllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} \\ \mathbf{4} & \mathbf{4} & \mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{3}+\mathbf{5} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} \\ \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{4} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\ \mathbf{7} & \mathbf{6} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{6} \ \mathbf{7}), (\mathbf{3} \ \mathbf{4} \ \mathbf{5})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5}\}\) | \(\left.\text{Rep(}D_7\right):\ \text{FR}^{5,0}_{4}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.\) | \(2\) |
| \(\mathbf{4}\) | \(2.\) | \(2\) |
| \(\mathbf{5}\) | \(2.\) | \(2\) |
| \(\mathbf{6}\) | \(2.64575\) | \(\sqrt{7}\) |
| \(\mathbf{7}\) | \(2.64575\) | \(\sqrt{7}\) |
| \(\mathcal{D}_{FP}^2\) | \(28.\) | \(28\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \hline 1 & 1 & 2 & 2 & 2 & \sqrt{7} & \sqrt{7} \\ 1 & 1 & 2 & 2 & 2 & -\sqrt{7} & -\sqrt{7} \\ 1 & 1 & \text{Root}\left[x^3+x^2-2 x-1,3\right] & \text{Root}\left[x^3+x^2-2 x-1,1\right] & \text{Root}\left[x^3+x^2-2 x-1,2\right] & 0 & 0 \\ 1 & 1 & \text{Root}\left[x^3+x^2-2 x-1,2\right] & \text{Root}\left[x^3+x^2-2 x-1,3\right] & \text{Root}\left[x^3+x^2-2 x-1,1\right] & 0 & 0 \\ 1 & 1 & \text{Root}\left[x^3+x^2-2 x-1,1\right] & \text{Root}\left[x^3+x^2-2 x-1,2\right] & \text{Root}\left[x^3+x^2-2 x-1,3\right] & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & i & -i \\ 1 & -1 & 0 & 0 & 0 & -i & i \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \hline 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 2.646 & 2.646 \\ 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & -2.646 & -2.646 \\ 1.000 & 1.000 & 1.247 & -1.802 & -0.4450 & 0 & 0 \\ 1.000 & 1.000 & -0.4450 & 1.247 & -1.802 & 0 & 0 \\ 1.000 & 1.000 & -1.802 & -0.4450 & 1.247 & 0 & 0 \\ 1.000 & -1.000 & 0 & 0 & 0 & 1.000 i & -1.000 i \\ 1.000 & -1.000 & 0 & 0 & 0 & -1.000 i & 1.000 i \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}\), form the adjoint subring \(\left.\text{Rep(}D_7\right):\ \text{FR}^{5,0}_{4}\) .
The upper central series is the following: \(\text{FR}^{7,2}_{12} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5} }{\supset} \left.\text{Rep(}D_7\right)\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{2}'\), 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
Download links for numeric data: