$\text{FR}^{8,6}_{5}$

Fusion Rules

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

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

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

The following particles form non-trivial sub fusion rings

Particles	SubRing
$\{\mathbf{1},\mathbf{2}\}$	$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$
$\{\mathbf{1},\mathbf{3},\mathbf{4}\}$	$\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\}$	$\mathbb{Z}_6:\ \text{FR}^{6,4}_{1}$

Quantum Dimensions

Particle	Numeric	Symbolic
$\mathbf{1}$	$1.$	$1$
$\mathbf{2}$	$1.$	$1$
$\mathbf{3}$	$1.$	$1$
$\mathbf{4}$	$1.$	$1$
$\mathbf{5}$	$1.$	$1$
$\mathbf{6}$	$1.$	$1$
$\mathbf{7}$	$1.73205$	$\sqrt{3}$
$\mathbf{8}$	$1.73205$	$\sqrt{3}$
$\mathcal{D}_{FP}^2$	$12.$	$12$

Characters

The symbolic character table is the following

$\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{4} & \mathbf{5} & \mathbf{3} & \mathbf{6} & \mathbf{2} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & \sqrt{3} & \sqrt{3} \\ 1 & 1 & 1 & 1 & 1 & 1 & -\sqrt{3} & -\sqrt{3} \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(1-i \sqrt{3}\right) & -1 & 0 & 0 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(1+i \sqrt{3}\right) & -1 & 0 & 0 \\ 1 & 1 & -1 & 1 & -1 & -1 & i \sqrt{3} & -i \sqrt{3} \\ 1 & 1 & -1 & 1 & -1 & -1 & -i \sqrt{3} & i \sqrt{3} \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 1 & 0 & 0 \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 1 & 0 & 0 \\ \hline \end{array}$

The numeric character table is the following

$\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{4} & \mathbf{5} & \mathbf{3} & \mathbf{6} & \mathbf{2} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.732 & 1.732 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & -1.732 & -1.732 \\ 1.000 & -0.5000-0.8660 i & 0.5000+0.8660 i & -0.5000+0.8660 i & 0.5000-0.8660 i & -1.000 & 0 & 0 \\ 1.000 & -0.5000+0.8660 i & 0.5000-0.8660 i & -0.5000-0.8660 i & 0.5000+0.8660 i & -1.000 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & 1.000 & -1.000 & -1.000 & 1.732 i & -1.732 i \\ 1.000 & 1.000 & -1.000 & 1.000 & -1.000 & -1.000 & -1.732 i & 1.732 i \\ 1.000 & -0.5000+0.8660 i & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.5000-0.8660 i & 1.000 & 0 & 0 \\ 1.000 & -0.5000-0.8660 i & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.5000+0.8660 i & 1.000 & 0 & 0 \\ \hline \end{array}$

Modular Data

This fusion ring does not have any matching $S$-and $T$-matrices.

Adjoint Subring

Particles $\mathbf{1}, \mathbf{3}, \mathbf{4}$, form the adjoint subring $\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}$ .

The upper central series is the following: $\text{FR}^{8,6}_{5} \underset{ \mathbf{1}, \mathbf{3}, \mathbf{4} }{\supset} \mathbb{Z}_3 \underset{ \mathbf{1} }{\supset} \text{Trivial}$

Universal grading

Each particle can be graded as follows: $\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{2}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{3}', \text{deg}(\mathbf{8}) = \mathbf{4}'$, where the degrees form the group $\mathbb{Z}_4$ with multiplication table:

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

Categorifications

Data

Download links for numeric data:

• Multiplication Table
• Quantum Dimensions
• Character Table