\(\text{Fib}\times \mathbb{Z}_3:\ \text{FR}^{6,4}_{5}\)
Fusion Rules
\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{6} & \mathbf{4} & \mathbf{5} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{4} \\ \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{1}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{6} \\ \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{6} & \mathbf{1}+\mathbf{4} \\ \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{2}+\mathbf{6} & \mathbf{1}+\mathbf{4} & \mathbf{3}+\mathbf{5} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.61803\) | \(\phi\) |
| \(\mathbf{5}\) | \(1.61803\) | \(\phi\) |
| \(\mathbf{6}\) | \(1.61803\) | \(\phi\) |
| \(\mathcal{D}_{FP}^2\) | \(10.8541\) | \(\frac{3 \sqrt{5}}{2}+\frac{15}{2}\) |
Here $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio.
Characters
The symbolic character table is the following
\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{4} \\ \hline 1 & 1 & 1 & \phi & \phi & \phi \\ 1 & 1 & 1 & -\phi^{-1} & -\phi^{-1} & -\phi^{-1} \\ 1 & e^{-\frac{1}{3} (2 i \pi )} & e^{\frac{1}{3} (2 i \pi )} & - e^{-\frac{1}{3} (2 i \pi )} \phi^{-1} & - e^{\frac{1}{3} (2 i \pi )} \phi^{-1} & -\phi^{-1} \\ 1 & e^{\frac{1}{3} (2 i \pi )} & e^{-\frac{1}{3} (2 i \pi )} & - e^{\frac{1}{3} (2 i \pi )} \phi^{-1} & -e^{-\frac{1}{3} (2 i \pi )} \phi^{-1} & -\phi^{-1} \\ 1 & e^{\frac{1}{3} (2 i \pi )} & e^{-\frac{1}{3} (2 i \pi )} & e^{\frac{1}{3} (2 i \pi )} \phi & e^{-\frac{1}{3} (2 i \pi )} \phi & \phi \\ 1 & e^{-\frac{1}{3} (2 i \pi )} & e^{\frac{1}{3} (2 i \pi )} & e^{-\frac{1}{3} (2 i \pi )} \phi & e^{\frac{1}{3} (2 i \pi )} \phi & \phi \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{4} \\ \hline 1.000 & 1.000 & 1.000 & 1.618 & 1.618 & 1.618 \\ 1.000 & 1.000 & 1.000 & -0.6180 & -0.6180 & -0.6180 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.3090+0.5352 i & 0.3090-0.5352 i & -0.6180 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.3090-0.5352 i & 0.3090+0.5352 i & -0.6180 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.809+1.401 i & -0.809-1.401 i & 1.618 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.809-1.401 i & -0.809+1.401 i & 1.618 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{\frac{3 \sqrt{5}}{2}+\frac{15}{2}}}\left(\begin{array}{cccccc} 1 & 1 & 1 & \phi & \phi & \phi \\ 1 & e^{-\frac{1}{3} (2 i \pi )} & e^{\frac{1}{3} (2 i \pi )} & \phi & e^{\frac{1}{3} (2 i \pi )} \phi & e^{-\frac{1}{3} (2 i \pi )} \phi \\ 1 & e^{\frac{1}{3} (2 i \pi )} & e^{-\frac{1}{3} (2 i \pi )} & \phi & e^{-\frac{1}{3} (2 i \pi )} \phi & e^{\frac{1}{3} (2 i \pi )} \phi \\ \phi & \phi & \phi & -1 & -1 & -1 \\ \phi & e^{-\frac{1}{3} (2 i \pi )} \phi & e^{\frac{1}{3} (2 i \pi )} \phi & -1 & e^{\frac{1}{3} (2 i \pi )} & e^{-\frac{1}{3} (2 i \pi )} \\ \phi & e^{\frac{1}{3} (2 i \pi )} \phi & e^{-\frac{1}{3} (2 i \pi )} \phi & -1 & e^{-\frac{1}{3} (2 i \pi )} & e^{\frac{1}{3} (2 i \pi )} \end{array}\right)\) $$ | \(\begin{array}{l}\left(0,-\frac{1}{3},-\frac{1}{3},\frac{2}{5},\frac{1}{15},\frac{1}{15}\right) \\\left(0,-\frac{1}{3},-\frac{1}{3},-\frac{2}{5},\frac{4}{15},\frac{4}{15}\right)\end{array}\) |
| \(\frac{1}{\sqrt{\frac{3 \sqrt{5}}{2}+\frac{15}{2}}}\left(\begin{array}{cccccc} 1 & 1 & 1 & \phi & \phi & \phi \\ 1 & e^{\frac{1}{3} (2 i \pi )} & e^{-\frac{1}{3} (2 i \pi )} & \phi & e^{-\frac{1}{3} (2 i \pi )} \phi & e^{\frac{1}{3} (2 i \pi )} \phi \\ 1 & e^{-\frac{1}{3} (2 i \pi )} & e^{\frac{1}{3} (2 i \pi )} & \phi & e^{\frac{1}{3} (2 i \pi )} \phi & e^{-\frac{1}{3} (2 i \pi )} \phi \\ \phi & \phi & \phi & -1 & -1 & -1 \\ \phi & e^{\frac{1}{3} (2 i \pi )} \phi & e^{-\frac{1}{3} (2 i \pi )} \phi & -1 & e^{-\frac{1}{3} (2 i \pi )} & e^{\frac{1}{3} (2 i \pi )} \\ \phi & e^{-\frac{1}{3} (2 i \pi )} \phi & e^{\frac{1}{3} (2 i \pi )} \phi & -1 & e^{\frac{1}{3} (2 i \pi )} & e^{-\frac{1}{3} (2 i \pi )} \end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{3},\frac{1}{3},\frac{2}{5},-\frac{4}{15},-\frac{4}{15}\right) \\\left(0,\frac{1}{3},\frac{1}{3},-\frac{2}{5},-\frac{1}{15},-\frac{1}{15}\right)\end{array}\) |
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{4}\), form the adjoint subring \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) .
The upper central series is the following: \(\text{Fib}\times \mathbb{Z}_3 \underset{ \mathbf{1}, \mathbf{4} }{\supset} \text{Fib}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{3}', \text{deg}(\mathbf{6}) = \mathbf{2}'\), where the degrees form the group \(\mathbb{Z}_3\) with multiplication table:
\[\begin{array}{|lll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' \\ \mathbf{2}' & \mathbf{3}' & \mathbf{1}' \\ \mathbf{3}' & \mathbf{1}' & \mathbf{2}' \\ \hline \end{array}\]Categorifications
Data
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