\(\mathbb{Z}_8:\ \text{FR}^{8,6}_{2}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{6} & \mathbf{8} & \mathbf{1} & \mathbf{2} & \mathbf{7} & \mathbf{4} & \mathbf{5} \\ \mathbf{4} & \mathbf{5} & \mathbf{1} & \mathbf{7} & \mathbf{8} & \mathbf{2} & \mathbf{6} & \mathbf{3} \\ \mathbf{5} & \mathbf{4} & \mathbf{2} & \mathbf{8} & \mathbf{7} & \mathbf{1} & \mathbf{3} & \mathbf{6} \\ \mathbf{6} & \mathbf{3} & \mathbf{7} & \mathbf{2} & \mathbf{1} & \mathbf{8} & \mathbf{5} & \mathbf{4} \\ \mathbf{7} & \mathbf{8} & \mathbf{4} & \mathbf{6} & \mathbf{3} & \mathbf{5} & \mathbf{2} & \mathbf{1} \\ \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{3} & \mathbf{6} & \mathbf{4} & \mathbf{1} & \mathbf{2} \\ \hline \end{array}\]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 |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{7},\mathbf{8}\}\) | \(\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}\) |
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
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{3} & \mathbf{4} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ 1 & 1 & i & -i & -1 & -1 & -i & i \\ 1 & 1 & -i & i & -1 & -1 & i & -i \\ 1 & -1 & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] & i & -i & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] \\ 1 & -1 & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] & -i & i & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] \\ 1 & -1 & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] & i & -i & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] \\ 1 & -1 & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] & -i & i & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{3} & \mathbf{4} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.000 & 1.000 & -1.000 & -1.000 \\ 1.000 & 1.000 & 1.000 i & -1.000 i & -1.000 & -1.000 & -1.000 i & 1.000 i \\ 1.000 & 1.000 & -1.000 i & 1.000 i & -1.000 & -1.000 & 1.000 i & -1.000 i \\ 1.000 & -1.000 & 0.7071-0.7071 i & 0.7071+0.7071 i & 1.000 i & -1.000 i & -0.7071-0.7071 i & -0.7071+0.7071 i \\ 1.000 & -1.000 & 0.7071+0.7071 i & 0.7071-0.7071 i & -1.000 i & 1.000 i & -0.7071+0.7071 i & -0.7071-0.7071 i \\ 1.000 & -1.000 & -0.7071+0.7071 i & -0.7071-0.7071 i & 1.000 i & -1.000 i & 0.7071+0.7071 i & 0.7071-0.7071 i \\ 1.000 & -1.000 & -0.7071-0.7071 i & -0.7071+0.7071 i & -1.000 i & 1.000 i & 0.7071-0.7071 i & 0.7071+0.7071 i \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{2 \sqrt{2}}\left(\begin{array}{cccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & -1 & -\frac{1+i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & -i & i \\ 1 & -1 & -\frac{1-i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & i & -i \\ 1 & -1 & \frac{1-i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & i & -i \\ 1 & -1 & \frac{1+i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & -i & i \\ 1 & 1 & -i & i & i & -i & -1 & -1 \\ 1 & 1 & i & -i & -i & i & -1 & -1 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,\frac{3}{16},\frac{3}{16},-\frac{5}{16},-\frac{5}{16},-\frac{1}{4},-\frac{1}{4}\right) \\\left(0,0,-\frac{5}{16},-\frac{5}{16},\frac{3}{16},\frac{3}{16},-\frac{1}{4},-\frac{1}{4}\right)\end{array}\) |
| \(\frac{1}{2 \sqrt{2}}\left(\begin{array}{cccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & -1 & -\frac{1-i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & i & -i \\ 1 & -1 & -\frac{1+i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & -i & i \\ 1 & -1 & \frac{1+i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & -i & i \\ 1 & -1 & \frac{1-i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & i & -i \\ 1 & 1 & i & -i & -i & i & -1 & -1 \\ 1 & 1 & -i & i & i & -i & -1 & -1 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,\frac{5}{16},\frac{5}{16},-\frac{3}{16},-\frac{3}{16},\frac{1}{4},\frac{1}{4}\right) \\\left(0,0,-\frac{3}{16},-\frac{3}{16},\frac{5}{16},\frac{5}{16},\frac{1}{4},\frac{1}{4}\right)\end{array}\) |
| \(\frac{1}{2 \sqrt{2}}\left(\begin{array}{cccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & -1 & \frac{1-i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & i & -i \\ 1 & -1 & \frac{1+i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & -i & i \\ 1 & -1 & -\frac{1+i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & -i & i \\ 1 & -1 & -\frac{1-i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & i & -i \\ 1 & 1 & i & -i & -i & i & -1 & -1 \\ 1 & 1 & -i & i & i & -i & -1 & -1 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,-\frac{7}{16},-\frac{7}{16},\frac{1}{16},\frac{1}{16},\frac{1}{4},\frac{1}{4}\right) \\\left(0,0,\frac{1}{16},\frac{1}{16},-\frac{7}{16},-\frac{7}{16},\frac{1}{4},\frac{1}{4}\right)\end{array}\) |
| \(\frac{1}{2 \sqrt{2}}\left(\begin{array}{cccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & -1 & \frac{1+i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & -i & i \\ 1 & -1 & \frac{1-i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & i & -i \\ 1 & -1 & -\frac{1-i}{\sqrt{2}} & -\frac{1+i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & i & -i \\ 1 & -1 & -\frac{1+i}{\sqrt{2}} & -\frac{1-i}{\sqrt{2}} & \frac{1-i}{\sqrt{2}} & \frac{1+i}{\sqrt{2}} & -i & i \\ 1 & 1 & -i & i & i & -i & -1 & -1 \\ 1 & 1 & i & -i & -i & i & -1 & -1 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,-\frac{1}{16},-\frac{1}{16},\frac{7}{16},\frac{7}{16},-\frac{1}{4},-\frac{1}{4}\right) \\\left(0,0,\frac{7}{16},\frac{7}{16},-\frac{1}{16},-\frac{1}{16},-\frac{1}{4},-\frac{1}{4}\right)\end{array}\) |
Adjoint Subring
The adjoint subring is the trivial ring.
The upper central series is the following: \(\mathbb{Z}_8 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}', \text{deg}(\mathbf{4}) = \mathbf{4}', \text{deg}(\mathbf{5}) = \mathbf{5}', \text{deg}(\mathbf{6}) = \mathbf{6}', \text{deg}(\mathbf{7}) = \mathbf{7}', \text{deg}(\mathbf{8}) = \mathbf{8}'\), where the degrees form the group \(\mathbb{Z}_8\) with multiplication table:
\[\begin{array}{|llllllll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' & \mathbf{5}' & \mathbf{6}' & \mathbf{7}' & \mathbf{8}' \\ \mathbf{2}' & \mathbf{1}' & \mathbf{6}' & \mathbf{5}' & \mathbf{4}' & \mathbf{3}' & \mathbf{8}' & \mathbf{7}' \\ \mathbf{3}' & \mathbf{6}' & \mathbf{8}' & \mathbf{1}' & \mathbf{2}' & \mathbf{7}' & \mathbf{4}' & \mathbf{5}' \\ \mathbf{4}' & \mathbf{5}' & \mathbf{1}' & \mathbf{7}' & \mathbf{8}' & \mathbf{2}' & \mathbf{6}' & \mathbf{3}' \\ \mathbf{5}' & \mathbf{4}' & \mathbf{2}' & \mathbf{8}' & \mathbf{7}' & \mathbf{1}' & \mathbf{3}' & \mathbf{6}' \\ \mathbf{6}' & \mathbf{3}' & \mathbf{7}' & \mathbf{2}' & \mathbf{1}' & \mathbf{8}' & \mathbf{5}' & \mathbf{4}' \\ \mathbf{7}' & \mathbf{8}' & \mathbf{4}' & \mathbf{6}' & \mathbf{3}' & \mathbf{5}' & \mathbf{2}' & \mathbf{1}' \\ \mathbf{8}' & \mathbf{7}' & \mathbf{5}' & \mathbf{3}' & \mathbf{6}' & \mathbf{4}' & \mathbf{1}' & \mathbf{2}' \\ \hline \end{array}\]Categorifications
Data
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