$\text{SU(2})_4:\ \text{FR}^{5,0}_{3}$

Fusion Rules

$\begin{array}{|lllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{5} & \mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{5} & \mathbf{1}+\mathbf{5} & \mathbf{3}+\mathbf{4} \\ \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{5} \\ \hline \end{array}$

The fusion rules are invariant under the group generated by the following permutations:

$\{(\mathbf{3} \ \mathbf{4})\}$

The following elements form non-trivial sub fusion rings

Elements	SubRing
$\{\mathbf{1},\mathbf{2}\}$	$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{5}\}$	$\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}$

Frobenius-Perron Dimensions

Particle	Numeric	Symbolic
$\mathbf{1}$	$1.$	$1$
$\mathbf{2}$	$1.$	$1$
$\mathbf{3}$	$1.73205$	$\sqrt{3}$
$\mathbf{4}$	$1.73205$	$\sqrt{3}$
$\mathbf{5}$	$2.$	$2$
$\mathcal{D}_{FP}^2$	$12.$	$12$

Characters

The symbolic character table is the following

$\begin{array}{|ccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & 0 & 0 & -1 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ 1 & -1 & 1 & -1 & 0 \\ 1 & -1 & -1 & 1 & 0 \\ \hline \end{array}$

The numeric character table is the following

$\begin{array}{|rrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline 1.000 & 1.000 & 1.732 & 1.732 & 2.000 \\ 1.000 & 1.000 & 0 & 0 & -1.000 \\ 1.000 & 1.000 & -1.732 & -1.732 & 2.000 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 \\ \hline \end{array}$

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

The matching $S$-matrices and twist factors are the following

$S$-matrix	Twist factors
$\frac{1}{2 \sqrt{3}}\left(\begin{array}{ccccc} 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ \sqrt{3} & -\sqrt{3} & \sqrt{3} & -\sqrt{3} & 0 \\ \sqrt{3} & -\sqrt{3} & -\sqrt{3} & \sqrt{3} & 0 \\ 2 & 2 & 0 & 0 & -2 \\\end{array}\right)$	$\begin{array}{l}\left(0,0,\frac{3}{8},-\frac{1}{8},-\frac{1}{3}\right) \\\left(0,0,-\frac{1}{8},\frac{3}{8},-\frac{1}{3}\right) \\\left(0,0,\frac{1}{8},-\frac{3}{8},\frac{1}{3}\right) \\\left(0,0,-\frac{3}{8},\frac{1}{8},\frac{1}{3}\right)\end{array}$
$\frac{1}{2 \sqrt{3}}\left(\begin{array}{ccccc} 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ \sqrt{3} & -\sqrt{3} & -\sqrt{3} & \sqrt{3} & 0 \\ \sqrt{3} & -\sqrt{3} & \sqrt{3} & -\sqrt{3} & 0 \\ 2 & 2 & 0 & 0 & -2 \\\end{array}\right)$	$\begin{array}{l}\left(0,0,-\frac{1}{8},\frac{3}{8},\frac{1}{3}\right) \\\left(0,0,\frac{3}{8},-\frac{1}{8},\frac{1}{3}\right) \\\left(0,0,-\frac{3}{8},\frac{1}{8},-\frac{1}{3}\right) \\\left(0,0,\frac{1}{8},-\frac{3}{8},-\frac{1}{3}\right)\end{array}$

Adjoint Subring

Elements $\mathbf{1}, \mathbf{2}, \mathbf{5}$, form the adjoint subring $\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}$ .

The upper central series is the following: $\text{SU(2})_4 \underset{ \mathbf{1}, \mathbf{2}, \mathbf{5} }{\supset} \left.\text{Rep(}D_3\right)$

Universal grading

Each particle can be graded as follows: $\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}'$, 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:

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