\(\text{Fib} \times \text{Fib}:\ \text{FR}^{4,0}_{5}\)
Fusion Rules
\[\begin{array}{|llll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \mathbf{2} & \mathbf{1}+\mathbf{2} & \mathbf{4} & \mathbf{3}+\mathbf{4} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{3} & \mathbf{2}+\mathbf{4} \\ \mathbf{4} & \mathbf{3}+\mathbf{4} & \mathbf{2}+\mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{3}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.61803\) | \(\phi\) |
| \(\mathbf{3}\) | \(1.61803\) | \(\phi\) |
| \(\mathbf{4}\) | \(2.61803\) | \(1 + \phi\) |
| \(\mathcal{D}_{FP}^2\) | \(13.0902\) | \(5 + 5\phi\) |
here \(\phi = \frac{1 + \sqrt{5}}{2}\) is the golden ratio.
Characters
The symbolic character table is the following
\[\begin{array}{|cccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \hline 1 & \phi & \phi & 1 + \phi \\ 1 & \phi^{-1} & \phi & -1 \\ 1 & \phi & \phi^{-1} & -1 \\ 1 & \phi^{-1} & \phi^{-1} & 1 + \phi^{-1} \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \hline 1.000 & 1.618 & 1.618 & 2.618 \\ 1.000 & -0.6180 & 1.618 & -1.000 \\ 1.000 & 1.618 & -0.6180 & -1.000 \\ 1.000 & -0.6180 & -0.6180 & 0.3820 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{5 + 5 \phi} }\left(\begin{array}{cccc} 1 & \phi & \phi & 1 + \phi \\ \phi & -1 & 1 + \phi & -\phi \\ \phi & 1 + \phi & -1 & -\phi \\ 1 + \phi & -\phi &-\phi & 1 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,-\frac{2}{5},\frac{2}{5},0\right) \\\left(0,\frac{2}{5},-\frac{2}{5},0\right) \\\left(0,\frac{2}{5},\frac{2}{5},-\frac{1}{5}\right) \\\left(0,-\frac{2}{5},-\frac{2}{5},\frac{1}{5}\right)\end{array}\) |
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
Download links for numeric data: