\(\text{FR}^{8,0}_{13}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{4} & \mathbf{4} & \mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{4} & \mathbf{3}+\mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{4}+\mathbf{6} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{6} & \mathbf{3}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{8} & \mathbf{7} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{7} \ \mathbf{8}), (\mathbf{3} \ \mathbf{5} \ \mathbf{6})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{4}\}\) | \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\}\) | \(\text{FR}^{6,0}_{7}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.\) | \(2\) |
| \(\mathbf{4}\) | \(2.\) | \(2\) |
| \(\mathbf{5}\) | \(2.\) | \(2\) |
| \(\mathbf{6}\) | \(2.\) | \(2\) |
| \(\mathbf{7}\) | \(3.\) | \(3\) |
| \(\mathbf{8}\) | \(3.\) | \(3\) |
| \(\mathcal{D}_{FP}^2\) | \(36.\) | \(36\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\ 1 & 1 & 2 & 2 & 2 & 2 & -3 & -3 \\ 1 & 1 & \text{Root}\left[x^3-3 x+1,1\right] & -1 & \text{Root}\left[x^3-3 x+1,2\right] & \text{Root}\left[x^3-3 x+1,3\right] & 0 & 0 \\ 1 & 1 & -1 & 2 & -1 & -1 & 0 & 0 \\ 1 & 1 & \text{Root}\left[x^3-3 x+1,2\right] & -1 & \text{Root}\left[x^3-3 x+1,3\right] & \text{Root}\left[x^3-3 x+1,1\right] & 0 & 0 \\ 1 & 1 & \text{Root}\left[x^3-3 x+1,3\right] & -1 & \text{Root}\left[x^3-3 x+1,1\right] & \text{Root}\left[x^3-3 x+1,2\right] & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 & 1 & -1 \\ 1 & -1 & 0 & 0 & 0 & 0 & -1 & 1 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 2.000 & 3.000 & 3.000 \\ 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 2.000 & -3.000 & -3.000 \\ 1.000 & 1.000 & -1.879 & -1.000 & 0.3473 & 1.532 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & 2.000 & -1.000 & -1.000 & 0 & 0 \\ 1.000 & 1.000 & 0.3473 & -1.000 & 1.532 & -1.879 & 0 & 0 \\ 1.000 & 1.000 & 1.532 & -1.000 & -1.879 & 0.3473 & 0 & 0 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 1.000 & -1.000 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & -1.000 & 1.000 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{6}\left(\begin{array}{cccccccc} 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\ 1 & 1 & 2 & 2 & 2 & 2 & -3 & -3 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 0 & 0 \\ 2 & 2 & -2 & 4 & -2 & -2 & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 0 & 0 \\ 3 & -3 & 0 & 0 & 0 & 0 & 3 & -3 \\ 3 & -3 & 0 & 0 & 0 & 0 & -3 & 3 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,-\frac{2}{9},0,\frac{4}{9},\frac{1}{9},0,\frac{1}{2}\right) \\\left(0,0,-\frac{2}{9},0,\frac{4}{9},\frac{1}{9},\frac{1}{2},0\right) \\\left(0,0,\frac{2}{9},0,-\frac{4}{9},-\frac{1}{9},\frac{1}{2},0\right) \\\left(0,0,\frac{2}{9},0,-\frac{4}{9},-\frac{1}{9},0,\frac{1}{2}\right)\end{array}\) |
| \(\frac{1}{6}\left(\begin{array}{cccccccc} 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\ 1 & 1 & 2 & 2 & 2 & 2 & -3 & -3 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 0 & 0 \\ 2 & 2 & -2 & 4 & -2 & -2 & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 0 & 0 \\ 3 & -3 & 0 & 0 & 0 & 0 & -3 & 3 \\ 3 & -3 & 0 & 0 & 0 & 0 & 3 & -3 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,\frac{2}{9},0,-\frac{4}{9},-\frac{1}{9},-\frac{1}{4},\frac{1}{4}\right) \\\left(0,0,\frac{2}{9},0,-\frac{4}{9},-\frac{1}{9},\frac{1}{4},-\frac{1}{4}\right) \\\left(0,0,-\frac{2}{9},0,\frac{4}{9},\frac{1}{9},-\frac{1}{4},\frac{1}{4}\right) \\\left(0,0,-\frac{2}{9},0,\frac{4}{9},\frac{1}{9},\frac{1}{4},-\frac{1}{4}\right)\end{array}\) |
| \(\frac{1}{6}\left(\begin{array}{cccccccc} 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\ 1 & 1 & 2 & 2 & 2 & 2 & -3 & -3 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 0 & 0 \\ 2 & 2 & -2 & 4 & -2 & -2 & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 0 & 0 \\ 3 & -3 & 0 & 0 & 0 & 0 & 3 & -3 \\ 3 & -3 & 0 & 0 & 0 & 0 & -3 & 3 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,\frac{4}{9},0,\frac{1}{9},-\frac{2}{9},0,\frac{1}{2}\right) \\\left(0,0,\frac{4}{9},0,\frac{1}{9},-\frac{2}{9},\frac{1}{2},0\right) \\\left(0,0,-\frac{4}{9},0,-\frac{1}{9},\frac{2}{9},\frac{1}{2},0\right) \\\left(0,0,-\frac{4}{9},0,-\frac{1}{9},\frac{2}{9},0,\frac{1}{2}\right)\end{array}\) |
| \(\frac{1}{6}\left(\begin{array}{cccccccc} 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\ 1 & 1 & 2 & 2 & 2 & 2 & -3 & -3 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 0 & 0 \\ 2 & 2 & -2 & 4 & -2 & -2 & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 0 & 0 \\ 3 & -3 & 0 & 0 & 0 & 0 & -3 & 3 \\ 3 & -3 & 0 & 0 & 0 & 0 & 3 & -3 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,-\frac{4}{9},0,-\frac{1}{9},\frac{2}{9},-\frac{1}{4},\frac{1}{4}\right) \\\left(0,0,-\frac{4}{9},0,-\frac{1}{9},\frac{2}{9},\frac{1}{4},-\frac{1}{4}\right) \\\left(0,0,\frac{4}{9},0,\frac{1}{9},-\frac{2}{9},-\frac{1}{4},\frac{1}{4}\right) \\\left(0,0,\frac{4}{9},0,\frac{1}{9},-\frac{2}{9},\frac{1}{4},-\frac{1}{4}\right)\end{array}\) |
| \(\frac{1}{6}\left(\begin{array}{cccccccc} 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\ 1 & 1 & 2 & 2 & 2 & 2 & -3 & -3 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 0 & 0 \\ 2 & 2 & -2 & 4 & -2 & -2 & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 0 & 0 \\ 3 & -3 & 0 & 0 & 0 & 0 & 3 & -3 \\ 3 & -3 & 0 & 0 & 0 & 0 & -3 & 3 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,\frac{1}{9},0,-\frac{2}{9},\frac{4}{9},0,\frac{1}{2}\right) \\\left(0,0,\frac{1}{9},0,-\frac{2}{9},\frac{4}{9},\frac{1}{2},0\right) \\\left(0,0,-\frac{1}{9},0,\frac{2}{9},-\frac{4}{9},\frac{1}{2},0\right) \\\left(0,0,-\frac{1}{9},0,\frac{2}{9},-\frac{4}{9},0,\frac{1}{2}\right)\end{array}\) |
| \(\frac{1}{6}\left(\begin{array}{cccccccc} 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\ 1 & 1 & 2 & 2 & 2 & 2 & -3 & -3 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 0 & 0 \\ 2 & 2 & -2 & 4 & -2 & -2 & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 0 & 0 \\ 2 & 2 & 6 \text{Root}\left[27 x^3-9 x+1,1\right] & -2 & 6 \text{Root}\left[27 x^3-9 x+1,2\right] & 6 \text{Root}\left[27 x^3-9 x+1,3\right] & 0 & 0 \\ 3 & -3 & 0 & 0 & 0 & 0 & -3 & 3 \\ 3 & -3 & 0 & 0 & 0 & 0 & 3 & -3 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,-\frac{1}{9},0,\frac{2}{9},-\frac{4}{9},-\frac{1}{4},\frac{1}{4}\right) \\\left(0,0,-\frac{1}{9},0,\frac{2}{9},-\frac{4}{9},\frac{1}{4},-\frac{1}{4}\right) \\\left(0,0,\frac{1}{9},0,-\frac{2}{9},\frac{4}{9},-\frac{1}{4},\frac{1}{4}\right) \\\left(0,0,\frac{1}{9},0,-\frac{2}{9},\frac{4}{9},\frac{1}{4},-\frac{1}{4}\right)\end{array}\) |
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6}\), form the adjoint subring \(\text{FR}^{6,0}_{7}\) .
The upper central series is the following: \(\text{FR}^{8,0}_{13} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6} }{\supset} \text{FR}^{6,0}_{7}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{2}', \text{deg}(\mathbf{8}) = \mathbf{2}'\), 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: