$\left.\text{TY(}\mathbb{Z}_8\right):\ \text{FR}^{9,6}_{2}$

Fusion Rules

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

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

$\{(\mathbf{3} \ \mathbf{6}) (\mathbf{4} \ \mathbf{5}), (\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6}) (\mathbf{7} \ \mathbf{8})\}$

The following elements form non-trivial sub fusion rings

Elements	SubRing
$\{\mathbf{1},\mathbf{2}\}$	$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{7},\mathbf{8}\}$	$\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6},\mathbf{7},\mathbf{8}\}$	$\mathbb{Z}_8:\ \text{FR}^{8,6}_{2}$

Frobenius-Perron Dimensions

Particle	Numeric	Symbolic
$\mathbf{1}$	$1.$	$1$
$\mathbf{2}$	$1.$	$1$
$\mathbf{3}$	$1.$	$1$
$\mathbf{4}$	$1.$	$1$
$\mathbf{5}$	$1.$	$1$
$\mathbf{6}$	$1.$	$1$
$\mathbf{7}$	$1.$	$1$
$\mathbf{8}$	$1.$	$1$
$\mathbf{9}$	$2.82843$	$2 \sqrt{2}$
$\mathcal{D}_{FP}^2$	$16.$	$16$

Characters

The symbolic character table is the following

$\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{4} & \mathbf{3} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 \sqrt{2} \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -2 \sqrt{2} \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 0 \\ 1 & 1 & i & -i & -1 & -1 & -i & i & 0 \\ 1 & 1 & -i & i & -1 & -1 & i & -i & 0 \\ 1 & -1 & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] & i & -i & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] & 0 \\ 1 & -1 & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] & -i & i & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] & 0 \\ 1 & -1 & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] & i & -i & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] & 0 \\ 1 & -1 & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] & -i & i & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] & 0 \\ \hline \end{array}$

The numeric character table is the following

$\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{4} & \mathbf{3} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 2.828 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & -2.828 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.000 & 1.000 & -1.000 & -1.000 & 0 \\ 1.000 & 1.000 & 1.000 i & -1.000 i & -1.000 & -1.000 & -1.000 i & 1.000 i & 0 \\ 1.000 & 1.000 & -1.000 i & 1.000 i & -1.000 & -1.000 & 1.000 i & -1.000 i & 0 \\ 1.000 & -1.000 & 0.7071-0.7071 i & 0.7071+0.7071 i & 1.000 i & -1.000 i & -0.7071-0.7071 i & -0.7071+0.7071 i & 0 \\ 1.000 & -1.000 & 0.7071+0.7071 i & 0.7071-0.7071 i & -1.000 i & 1.000 i & -0.7071+0.7071 i & -0.7071-0.7071 i & 0 \\ 1.000 & -1.000 & -0.7071+0.7071 i & -0.7071-0.7071 i & 1.000 i & -1.000 i & 0.7071+0.7071 i & 0.7071-0.7071 i & 0 \\ 1.000 & -1.000 & -0.7071-0.7071 i & -0.7071+0.7071 i & -1.000 i & 1.000 i & 0.7071-0.7071 i & 0.7071+0.7071 i & 0 \\ \hline \end{array}$

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

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

Adjoint Subring

Elements $\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6}, \mathbf{7}, \mathbf{8}$, form the adjoint subring $\mathbb{Z}_8:\ \text{FR}^{8,6}_{2}$ .

The upper central series is the following: $\left.\text{TY(}\mathbb{Z}_8\right) \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6}, \mathbf{7}, \mathbf{8} }{\supset} \mathbb{Z}_8 \underset{ \mathbf{1} }{\supset} \text{Trivial}$

Universal grading

Each particle can be graded as follows: $\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{1}', \text{deg}(\mathbf{8}) = \mathbf{1}', \text{deg}(\mathbf{9}) = \mathbf{2}'$, 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