\(\left.\text{TY(}\mathbb{Z}_8\right):\ \text{FR}^{9,6}_{2}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{3} & \mathbf{6} & \mathbf{8} & \mathbf{1} & \mathbf{2} & \mathbf{7} & \mathbf{4} & \mathbf{5} & \mathbf{9} \\ \mathbf{4} & \mathbf{5} & \mathbf{1} & \mathbf{7} & \mathbf{8} & \mathbf{2} & \mathbf{6} & \mathbf{3} & \mathbf{9} \\ \mathbf{5} & \mathbf{4} & \mathbf{2} & \mathbf{8} & \mathbf{7} & \mathbf{1} & \mathbf{3} & \mathbf{6} & \mathbf{9} \\ \mathbf{6} & \mathbf{3} & \mathbf{7} & \mathbf{2} & \mathbf{1} & \mathbf{8} & \mathbf{5} & \mathbf{4} & \mathbf{9} \\ \mathbf{7} & \mathbf{8} & \mathbf{4} & \mathbf{6} & \mathbf{3} & \mathbf{5} & \mathbf{2} & \mathbf{1} & \mathbf{9} \\ \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{3} & \mathbf{6} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{3} \ \mathbf{6}) (\mathbf{4} \ \mathbf{5}), (\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6}) (\mathbf{7} \ \mathbf{8})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{7},\mathbf{8}\}\) | \(\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6},\mathbf{7},\mathbf{8}\}\) | \(\mathbb{Z}_8:\ \text{FR}^{8,6}_{2}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(1.\) | \(1\) |
| \(\mathbf{6}\) | \(1.\) | \(1\) |
| \(\mathbf{7}\) | \(1.\) | \(1\) |
| \(\mathbf{8}\) | \(1.\) | \(1\) |
| \(\mathbf{9}\) | \(2.82843\) | \(2 \sqrt{2}\) |
| \(\mathcal{D}_{FP}^2\) | \(16.\) | \(16\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{4} & \mathbf{3} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 \sqrt{2} \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -2 \sqrt{2} \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 0 \\ 1 & 1 & i & -i & -1 & -1 & -i & i & 0 \\ 1 & 1 & -i & i & -1 & -1 & i & -i & 0 \\ 1 & -1 & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] & i & -i & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] & 0 \\ 1 & -1 & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] & -i & i & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] & 0 \\ 1 & -1 & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] & i & -i & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] & 0 \\ 1 & -1 & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] & -i & i & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{4} & \mathbf{3} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 2.828 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & -2.828 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.000 & 1.000 & -1.000 & -1.000 & 0 \\ 1.000 & 1.000 & 1.000 i & -1.000 i & -1.000 & -1.000 & -1.000 i & 1.000 i & 0 \\ 1.000 & 1.000 & -1.000 i & 1.000 i & -1.000 & -1.000 & 1.000 i & -1.000 i & 0 \\ 1.000 & -1.000 & 0.7071-0.7071 i & 0.7071+0.7071 i & 1.000 i & -1.000 i & -0.7071-0.7071 i & -0.7071+0.7071 i & 0 \\ 1.000 & -1.000 & 0.7071+0.7071 i & 0.7071-0.7071 i & -1.000 i & 1.000 i & -0.7071+0.7071 i & -0.7071-0.7071 i & 0 \\ 1.000 & -1.000 & -0.7071+0.7071 i & -0.7071-0.7071 i & 1.000 i & -1.000 i & 0.7071+0.7071 i & 0.7071-0.7071 i & 0 \\ 1.000 & -1.000 & -0.7071-0.7071 i & -0.7071+0.7071 i & -1.000 i & 1.000 i & 0.7071-0.7071 i & 0.7071+0.7071 i & 0 \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6}, \mathbf{7}, \mathbf{8}\), form the adjoint subring \(\mathbb{Z}_8:\ \text{FR}^{8,6}_{2}\) .
The upper central series is the following: \(\left.\text{TY(}\mathbb{Z}_8\right) \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6}, \mathbf{7}, \mathbf{8} }{\supset} \mathbb{Z}_8 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{1}', \text{deg}(\mathbf{8}) = \mathbf{1}', \text{deg}(\mathbf{9}) = \mathbf{2}'\), 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: