\(\text{Fib$\times $Fib$\times $}\mathbb{Z}_2:\ \text{FR}^{8,0}_{7}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{6} & \mathbf{1}+\mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{2}+\mathbf{3} & \mathbf{5}+\mathbf{8} & \mathbf{4}+\mathbf{7} \\ \mathbf{4} & \mathbf{5} & \mathbf{8} & \mathbf{1}+\mathbf{5} & \mathbf{2}+\mathbf{4} & \mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{7} \\ \mathbf{5} & \mathbf{4} & \mathbf{7} & \mathbf{2}+\mathbf{4} & \mathbf{1}+\mathbf{5} & \mathbf{8} & \mathbf{3}+\mathbf{7} & \mathbf{6}+\mathbf{8} \\ \mathbf{6} & \mathbf{3} & \mathbf{2}+\mathbf{3} & \mathbf{7} & \mathbf{8} & \mathbf{1}+\mathbf{6} & \mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{5}+\mathbf{8} & \mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{7} & \mathbf{4}+\mathbf{7} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7} \\ \mathbf{8} & \mathbf{7} & \mathbf{4}+\mathbf{7} & \mathbf{3}+\mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{5}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{6}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{6}\}\) | \(\text{SU(2})_3:\ \text{FR}^{4,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{4},\mathbf{5}\}\) | \(\text{SU(2})_3:\ \text{FR}^{4,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{5},\mathbf{6},\mathbf{8}\}\) | \(\text{Fib$\times $Fib}:\ \text{FR}^{4,0}_{5}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{4}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{5}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{6}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{7}\) | \(2.61803\) | \(\frac{1}{2} \left(3+\sqrt{5}\right)\) |
| \(\mathbf{8}\) | \(2.61803\) | \(\frac{1}{2} \left(3+\sqrt{5}\right)\) |
| \(\mathcal{D}_{FP}^2\) | \(26.1803\) | \(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{3} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & -1 & -1 \\ 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -1 & -1 \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(-3-\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 1 & -1 \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 1 & -1 \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(\sqrt{5}-3\right) & \frac{1}{2} \left(3-\sqrt{5}\right) \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{3} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.618 & 1.618 & 1.618 & 1.618 & 2.618 & 2.618 \\ 1.000 & 1.000 & -0.6180 & 1.618 & 1.618 & -0.6180 & -1.000 & -1.000 \\ 1.000 & 1.000 & 1.618 & -0.6180 & -0.6180 & 1.618 & -1.000 & -1.000 \\ 1.000 & 1.000 & -0.6180 & -0.6180 & -0.6180 & -0.6180 & 0.3820 & 0.3820 \\ 1.000 & -1.000 & 1.618 & 1.618 & -1.618 & -1.618 & -2.618 & 2.618 \\ 1.000 & -1.000 & 1.618 & -0.6180 & 0.6180 & -1.618 & 1.000 & -1.000 \\ 1.000 & -1.000 & -0.6180 & 1.618 & -1.618 & 0.6180 & 1.000 & -1.000 \\ 1.000 & -1.000 & -0.6180 & -0.6180 & 0.6180 & 0.6180 & -0.3820 & 0.3820 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}\left(\begin{array}{cccccccc} \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,3\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,3\right] & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] \\ \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,3\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,2\right] & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,1\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] \\ \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,3\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,1\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,2\right] & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} \\ \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,1\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,3\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,2\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} \\ \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,2\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,2\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} \\ \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,2\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,2\right] & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} \\ \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,1\right] & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,2\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,3\right] \\ \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,4\right] & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & -\sqrt{\frac{1}{10} \left(2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\right)} & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,3\right] & \sqrt{2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[100 x^4-30 x^2+1,3\right] \\\end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{4},-\frac{3}{20},-\frac{7}{20},\frac{2}{5},-\frac{2}{5},\frac{1}{4},0\right) \\\left(0,\frac{1}{4},-\frac{7}{20},-\frac{3}{20},-\frac{2}{5},\frac{2}{5},\frac{1}{4},0\right) \\\left(0,-\frac{1}{4},\frac{7}{20},\frac{7}{20},-\frac{2}{5},-\frac{2}{5},-\frac{1}{20},\frac{1}{5}\right) \\\left(0,-\frac{1}{4},\frac{7}{20},\frac{3}{20},\frac{2}{5},-\frac{2}{5},-\frac{1}{4},0\right) \\\left(0,-\frac{1}{4},\frac{3}{20},\frac{7}{20},-\frac{2}{5},\frac{2}{5},-\frac{1}{4},0\right) \\\left(0,\frac{1}{4},-\frac{3}{20},-\frac{3}{20},-\frac{2}{5},-\frac{2}{5},\frac{9}{20},\frac{1}{5}\right) \\\left(0,-\frac{1}{4},\frac{3}{20},\frac{3}{20},\frac{2}{5},\frac{2}{5},-\frac{9}{20},-\frac{1}{5}\right) \\\left(0,\frac{1}{4},-\frac{7}{20},-\frac{7}{20},\frac{2}{5},\frac{2}{5},\frac{1}{20},-\frac{1}{5}\right)\end{array}\) |
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{5}, \mathbf{6}, \mathbf{8}\), form the adjoint subring \(\text{Fib$\times $Fib}:\ \text{FR}^{4,0}_{5}\) .
The upper central series is the following: \(\text{Fib$\times $Fib$\times $}\mathbb{Z}_2 \underset{ \mathbf{1}, \mathbf{5}, \mathbf{6}, \mathbf{8} }{\supset} \text{Fib$\times $Fib}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{2}', \text{deg}(\mathbf{8}) = \mathbf{1}'\), 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: