\(\left.\text{Fib$\times $Rep(}D_3\right):\ \text{FR}^{6,0}_{5}\)
Fusion Rules
\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{4} & \mathbf{2}+\mathbf{3} & \mathbf{6} & \mathbf{5}+\mathbf{6} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{3} & \mathbf{1}+\mathbf{4} & \mathbf{6} & \mathbf{5}+\mathbf{6} \\ \mathbf{5} & \mathbf{5} & \mathbf{6} & \mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{6} \\ \mathbf{6} & \mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \hline \end{array}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{5}\}\) | \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\text{SU(2})_3:\ \text{FR}^{4,0}_{2}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{4}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{5}\) | \(2.\) | \(2\) |
| \(\mathbf{6}\) | \(3.23607\) | \(1+\sqrt{5}\) |
| \(\mathcal{D}_{FP}^2\) | \(21.7082\) | \(6+\frac{3}{2} \left(1+\sqrt{5}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} \\ \hline 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 2 & 1+\sqrt{5} \\ 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 2 & 1-\sqrt{5} \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) \\ 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 0 & 0 \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.618 & 1.618 & 2.000 & 3.236 \\ 1.000 & 1.000 & 1.618 & 1.618 & -1.000 & -1.618 \\ 1.000 & 1.000 & -0.6180 & -0.6180 & 2.000 & -1.236 \\ 1.000 & 1.000 & -0.6180 & -0.6180 & -1.000 & 0.6180 \\ 1.000 & -1.000 & 1.618 & -1.618 & 0 & 0 \\ 1.000 & -1.000 & -0.6180 & 0.6180 & 0 & 0 \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
The adjoint subring is the ring itself.
The upper central series is trivial.
Universal grading
This fusion ring allows only the trivial grading.
Categorifications
Data
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