$\text{FR}^{8,4}_{11}$

Fusion Rules

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

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

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

The following particles form non-trivial sub fusion rings

Particles	SubRing
$\{\mathbf{1},\mathbf{2}\}$	$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}$	$\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{7},\mathbf{8}\}$	$\text{Pseudo PSU(2})_6:\ \text{FR}^{4,2}_{4}$

Quantum Dimensions

Particle	Numeric	Symbolic
$\mathbf{1}$	$1.$	$1$
$\mathbf{2}$	$1.$	$1$
$\mathbf{3}$	$1.$	$1$
$\mathbf{4}$	$1.$	$1$
$\mathbf{5}$	$2.41421$	$1+\sqrt{2}$
$\mathbf{6}$	$2.41421$	$1+\sqrt{2}$
$\mathbf{7}$	$2.41421$	$1+\sqrt{2}$
$\mathbf{8}$	$2.41421$	$1+\sqrt{2}$
$\mathcal{D}_{FP}^2$	$27.3137$	$4+4 \left(1+\sqrt{2}\right)^2$

Characters

The symbolic character table is the following

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

The numeric character table is the following

$\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.414 & 2.414 & 2.414 & 2.414 \\ 1.000 & 1.000 & 1.000 & 1.000 & -0.4142 & -0.4142 & -0.4142 & -0.4142 \\ 1.000 & 1.000 & -1.000 & -1.000 & -2.414 & 2.414 & 2.414 & -2.414 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0.4142 & -0.4142 & -0.4142 & 0.4142 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 1.000 & 1.000 i & -1.000 i & -1.000 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 1.000 & -1.000 i & 1.000 i & -1.000 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & -1.000 & 1.000 i & -1.000 i & 1.000 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & -1.000 & -1.000 i & 1.000 i & 1.000 \\ \hline \end{array}$

Modular Data

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

Adjoint Subring

Particles $\mathbf{1}, \mathbf{2}, \mathbf{7}, \mathbf{8}$, form the adjoint subring $\text{Pseudo PSU(2})_6:\ \text{FR}^{4,2}_{4}$ .

The upper central series is the following: $\text{FR}^{8,4}_{11} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{7}, \mathbf{8} }{\supset} \text{Pseudo PSU(2})_6$

Universal grading

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