\(\text{SU(2})_3:\ \text{FR}^{4,0}_{2}\)
This fusion ring is isomorphic to \(\mathbb{Z}_2 \times \text{PSU}(2)_3 \cong \mathbb{Z}_2 \times \text{Fib}\).
Fusion Rules
\[\begin{array}{|llll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{4} & \mathbf{2}+\mathbf{3} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{3} & \mathbf{1}+\mathbf{4} \\ \hline \end{array}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.61803\) | \(\phi\) |
| \(\mathbf{4}\) | \(1.61803\) | \(\phi\) |
| \(\mathcal{D}_{FP}^2\) | \(7.23607\) | \(4 + 2 \phi\) |
Here \(\phi= \frac{1}{2} \left(1+\sqrt{5}\right)\) stands for the golden ratio.
Characters
The symbolic character table is the following
\[\begin{array}{|cccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} \\ \hline 1 & 1 & \phi & \phi \\ 1 & 1 & -\phi^{-1} & -\phi^{-1} \\ 1 & -1 & \phi & -\phi \\ 1 & -1 & -\phi^{-1} & \phi^{-1} \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} \\ \hline 1.000 & 1.000 & 1.618 & 1.618 \\ 1.000 & 1.000 & -0.6180 & -0.6180 \\ 1.000 & -1.000 & 1.618 & -1.618 \\ 1.000 & -1.000 & -0.6180 & 0.6180 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{4 + 2 \phi }}\left(\begin{array}{cccc} 1 & 1 & \phi & \phi \\ 1 & -1 &-\phi & \phi \\ \phi & -\phi & 1 & -1 \\ \phi & \phi & -1 & -1 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{4},-\frac{3}{20},-\frac{2}{5}\right) \\\left(0,\frac{1}{4},-\frac{7}{20},\frac{2}{5}\right) \\\left(0,-\frac{1}{4},\frac{7}{20},-\frac{2}{5}\right) \\\left(0,-\frac{1}{4},\frac{3}{20},\frac{2}{5}\right)\end{array}\) |
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: \(\text{SU(2})_3 \underset{ \mathbf{1}, \mathbf{4} }{\supset} \text{Fib}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{1}'\), 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: