$\text{FR}^{9,2}_{3}$

Fusion Rules

$\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{5} & \mathbf{6} & \mathbf{5} & \mathbf{6} & \mathbf{1}+\mathbf{3} & \mathbf{2}+\mathbf{4} & \mathbf{9} & \mathbf{9} & \mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{5} & \mathbf{6} & \mathbf{5} & \mathbf{2}+\mathbf{4} & \mathbf{1}+\mathbf{3} & \mathbf{9} & \mathbf{9} & \mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{7} & \mathbf{9} & \mathbf{9} & \mathbf{2}+\mathbf{3} & \mathbf{1}+\mathbf{4} & \mathbf{5}+\mathbf{6} \\ \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{9} & \mathbf{1}+\mathbf{4} & \mathbf{2}+\mathbf{3} & \mathbf{5}+\mathbf{6} \\ \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} \\ \hline \end{array}$

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

$\{(\mathbf{5} \ \mathbf{6}), (\mathbf{7} \ \mathbf{8})\}$

The following elements form non-trivial sub fusion rings

Elements	SubRing
$\{\mathbf{1},\mathbf{2}\}$	$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$
$\{\mathbf{1},\mathbf{3}\}$	$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$
$\{\mathbf{1},\mathbf{4}\}$	$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$
$\{\mathbf{1},\mathbf{3},\mathbf{5}\}$	$\text{Ising}:\ \text{FR}^{3,0}_{1}$
$\{\mathbf{1},\mathbf{3},\mathbf{6}\}$	$\text{Ising}:\ \text{FR}^{3,0}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}$	$\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{9}\}$	$\left.\text{Rep(}D_4\right):\ \text{FR}^{5,0}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{7},\mathbf{8}\}$	$\text{FR}^{6,2}_{3}$
$\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\}$	\(\mathbb{Z}_2\text{$\times$Ising}:\ \text{FR}^{6,0}_{1}\)

Frobenius-Perron Dimensions

Particle	Numeric	Symbolic
$\mathbf{1}$	$1.$	$1$
$\mathbf{2}$	$1.$	$1$
$\mathbf{3}$	$1.$	$1$
$\mathbf{4}$	$1.$	$1$
$\mathbf{5}$	$1.41421$	$\sqrt{2}$
$\mathbf{6}$	$1.41421$	$\sqrt{2}$
$\mathbf{7}$	$1.41421$	$\sqrt{2}$
$\mathbf{8}$	$1.41421$	$\sqrt{2}$
$\mathbf{9}$	$2.$	$2$
$\mathcal{D}_{FP}^2$	$16.$	$16$

Characters

The symbolic character table is the following

$\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & \sqrt{2} & \sqrt{2} & \sqrt{2} & \sqrt{2} & 2 \\ 1 & 1 & 1 & 1 & \sqrt{2} & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & -2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & -\sqrt{2} & \sqrt{2} & \sqrt{2} & -2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & 2 \\ 1 & -1 & 1 & -1 & -\sqrt{2} & \sqrt{2} & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & \sqrt{2} & -\sqrt{2} & 0 & 0 & 0 \\ 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & -1 & 1 & 0 & 0 & i \sqrt{2} & -i \sqrt{2} & 0 \\ 1 & -1 & -1 & 1 & 0 & 0 & -i \sqrt{2} & i \sqrt{2} & 0 \\ \hline \end{array}$

The numeric character table is the following

$\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.414 & 1.414 & 1.414 & 1.414 & 2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.414 & 1.414 & -1.414 & -1.414 & -2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.414 & -1.414 & 1.414 & 1.414 & -2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.414 & -1.414 & -1.414 & -1.414 & 2.000 \\ 1.000 & -1.000 & 1.000 & -1.000 & -1.414 & 1.414 & 0 & 0 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & 1.414 & -1.414 & 0 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 & 1.414 i & -1.414 i & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 & -1.414 i & 1.414 i & 0 \\ \hline \end{array}$

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

This fusion ring does not provide any representations of $SL_2(\mathbb{Z}).$

Adjoint Subring

Elements $\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}$, form the adjoint subring $\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}$ .

The upper central series is the following: $\text{FR}^{9,2}_{3} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4} }{\supset} \mathbb{Z}_2\times \mathbb{Z}_2 \underset{ \mathbf{1} }{\supset} \text{Trivial}$

Universal grading

Each particle can be graded as follows: $\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{2}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{3}', \text{deg}(\mathbf{8}) = \mathbf{3}', \text{deg}(\mathbf{9}) = \mathbf{4}'$, where the degrees form the group $\mathbb{Z}_2\times \mathbb{Z}_2$ with multiplication table:

$\begin{array}{|llll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' \\ \mathbf{2}' & \mathbf{1}' & \mathbf{4}' & \mathbf{3}' \\ \mathbf{3}' & \mathbf{4}' & \mathbf{1}' & \mathbf{2}' \\ \mathbf{4}' & \mathbf{3}' & \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}$

Categorifications

Data

Download links for numeric data:

• Multiplication Table
• Quantum Dimensions
• Character Table