$\text{Rep}(SD_{16}):\ \text{FR}^{7,2}_{4}$

$SD_{16}$ GAP ID: [16,8]

Fusion Rules

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

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

$\{(\mathbf{2} \ \mathbf{3}), (\mathbf{6} \ \mathbf{7})\}$

The following particles form non-trivial sub fusion rings

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

Quantum Dimensions

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

Characters

The symbolic character table is the following

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

The numeric character table is the following

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

Modular Data

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

Adjoint Subring

Particles $\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}$, form the adjoint subring $\left.\text{Rep(}D_4\right):\ \text{FR}^{5,0}_{1}$ .

The upper central series is the following: $\text{FR}^{7,2}_{4} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5} }{\supset} \left.\text{Rep(}D_4\right) \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4} }{\supset} \mathbb{Z}_2\times \mathbb{Z}_2 \underset{ \mathbf{1} }{\supset} \text{Trivial}$

Universal grading

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