\(\text{Fib$\times$}\text{PSU}(2)_6:\ \text{FR}^{8,0}_{16}\)

Fusion Rules

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

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

$\{(\mathbf{5} \ \mathbf{6}) (\mathbf{7} \ \mathbf{8})\}$

The following elements form non-trivial sub fusion rings

Elements	SubRing
$\{\mathbf{1},\mathbf{2}\}$	$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$
$\{\mathbf{1},\mathbf{4}\}$	$\text{Fib}:\ \text{FR}^{2,0}_{2}$
$\{\mathbf{1},\mathbf{2},\mathbf{5},\mathbf{6}\}$	$\text{PSU(2})_6:\ \text{FR}^{4,0}_{4}$
$\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}$	$\text{SU(2})_3:\ \text{FR}^{4,0}_{2}$

Frobenius-Perron Dimensions

Particle	Numeric	Symbolic
$\mathbf{1}$	$1.$	$1$
$\mathbf{2}$	$1.$	$1$
$\mathbf{3}$	$1.61803$	$\frac{1}{2} \left(1+\sqrt{5}\right)$
$\mathbf{4}$	$1.61803$	$\frac{1}{2} \left(1+\sqrt{5}\right)$
$\mathbf{5}$	$2.41421$	$1+\sqrt{2}$
$\mathbf{6}$	$2.41421$	$1+\sqrt{2}$
$\mathbf{7}$	$3.90628$	$\text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right]$
$\mathbf{8}$	$3.90628$	$\text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right]$
$\mathcal{D}_{FP}^2$	$49.411$	$2 \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right]^2+2+2 \left(1+\sqrt{2}\right)^2+\frac{1}{2} \left(1+\sqrt{5}\right)^2$

Characters

The symbolic character table is the following

$\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 1+\sqrt{2} & 1+\sqrt{2} & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right] & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right] \\ 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 1-\sqrt{2} & 1-\sqrt{2} & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,2\right] & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,2\right] \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 1+\sqrt{2} & 1+\sqrt{2} & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,1\right] & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,1\right] \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 1-\sqrt{2} & 1-\sqrt{2} & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,3\right] & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,3\right] \\ 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 1 & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & -1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 1 & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & -1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) \\ \hline \end{array}$

The numeric character table is the following

$\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.618 & 1.618 & 2.414 & 2.414 & 3.906 & 3.906 \\ 1.000 & 1.000 & 1.618 & 1.618 & -0.4142 & -0.4142 & -0.6702 & -0.6702 \\ 1.000 & 1.000 & -0.6180 & -0.6180 & 2.414 & 2.414 & -1.492 & -1.492 \\ 1.000 & 1.000 & -0.6180 & -0.6180 & -0.4142 & -0.4142 & 0.2560 & 0.2560 \\ 1.000 & -1.000 & 1.618 & -1.618 & 1.000 & -1.000 & -1.618 & 1.618 \\ 1.000 & -1.000 & 1.618 & -1.618 & -1.000 & 1.000 & 1.618 & -1.618 \\ 1.000 & -1.000 & -0.6180 & 0.6180 & 1.000 & -1.000 & 0.6180 & -0.6180 \\ 1.000 & -1.000 & -0.6180 & 0.6180 & -1.000 & 1.000 & -0.6180 & 0.6180 \\ \hline \end{array}$

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

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

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

Data

Download links for numeric data:

• Multiplication Table
• Quantum Dimensions
• Character Table