$\text{FR}^{7,2}_{13}$

Fusion Rules

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

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

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

The following particles form non-trivial sub fusion rings

Particles	SubRing
$\{\mathbf{1},\mathbf{2}\}$	$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$
$\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5}\}$	$\left.\text{Rep(}D_7\right):\ \text{FR}^{5,0}_{4}$

Quantum Dimensions

Particle	Numeric	Symbolic
$\mathbf{1}$	$1.$	$1$
$\mathbf{2}$	$1.$	$1$
$\mathbf{3}$	$2.$	$2$
$\mathbf{4}$	$2.$	$2$
$\mathbf{5}$	$2.$	$2$
$\mathbf{6}$	$3.82843$	$1+2 \sqrt{2}$
$\mathbf{7}$	$3.82843$	$1+2 \sqrt{2}$
$\mathcal{D}_{FP}^2$	$43.3137$	$14+2 \left(1+2 \sqrt{2}\right)^2$

Characters

The symbolic character table is the following

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

The numeric character table is the following

$\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \hline 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 3.828 & 3.828 \\ 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & -1.828 & -1.828 \\ 1.000 & 1.000 & 1.247 & -1.802 & -0.4450 & 0 & 0 \\ 1.000 & 1.000 & -0.4450 & 1.247 & -1.802 & 0 & 0 \\ 1.000 & 1.000 & -1.802 & -0.4450 & 1.247 & 0 & 0 \\ 1.000 & -1.000 & 0 & 0 & 0 & 1.000 i & -1.000 i \\ 1.000 & -1.000 & 0 & 0 & 0 & -1.000 i & 1.000 i \\ \hline \end{array}$

Modular Data

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

Adjoint Subring

The adjoint subring is the ring itself.

The upper central series is trivial.

Universal grading

This fusion ring allows only the trivial grading.

Categorifications

This fusion ring has no categorifications because of the $d$-number criterion.

Data

Download links for numeric data:

• Multiplication Table
• Quantum Dimensions
• Character Table