$\text{Fib}\times \mathbb{Z}_3:\ \text{FR}^{6,4}_{5}$

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 elements form non-trivial sub fusion rings

Elements	SubRing
$\{\mathbf{1},\mathbf{4}\}$	$\text{Fib}:\ \text{FR}^{2,0}_{2}$
$\{\mathbf{1},\mathbf{2},\mathbf{3}\}$	$\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}$

Frobenius-Perron 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

$\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{4} \\ \hline 1.000 & 1.000 & 1.000 & 1.618 & 1.618 & 1.618 \\ 1.000 & 1.000 & 1.000 & -0.6180 & -0.6180 & -0.6180 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.3090+0.5352 i & 0.3090-0.5352 i & -0.6180 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.3090-0.5352 i & 0.3090+0.5352 i & -0.6180 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.809+1.401 i & -0.809-1.401 i & 1.618 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.809-1.401 i & -0.809+1.401 i & 1.618 \\ \hline \end{array}$

Representations of $SL_2(\mathbb{Z})$

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

Elements $\mathbf{1}, \mathbf{4}$, 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

Download links for numeric data:

• Multiplication Table
• Quantum Dimensions
• Character Table
• S-Matrix 1 , Twist Factors1
• S-Matrix 2 , Twist Factors2