\(\left.\text{Fib$\times $Rep(}D_3\right):\ \text{FR}^{6,0}_{5}\)

Fusion Rules

\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{4} & \mathbf{2}+\mathbf{3} & \mathbf{6} & \mathbf{5}+\mathbf{6} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{3} & \mathbf{1}+\mathbf{4} & \mathbf{6} & \mathbf{5}+\mathbf{6} \\ \mathbf{5} & \mathbf{5} & \mathbf{6} & \mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{6} \\ \mathbf{6} & \mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \hline \end{array}\]

The following particles form non-trivial sub fusion rings

Particles SubRing
\(\{\mathbf{1},\mathbf{2}\}\) \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\)
\(\{\mathbf{1},\mathbf{4}\}\) \(\text{Fib}:\ \text{FR}^{2,0}_{2}\)
\(\{\mathbf{1},\mathbf{2},\mathbf{5}\}\) \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\)
\(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) \(\text{SU(2})_3:\ \text{FR}^{4,0}_{2}\)

Quantum Dimensions

Particle Numeric Symbolic
\(\mathbf{1}\) \(1.\) \(1\)
\(\mathbf{2}\) \(1.\) \(1\)
\(\mathbf{3}\) \(1.61803\) \(\frac{1}{2} \left(1+\sqrt{5}\right)\)
\(\mathbf{4}\) \(1.61803\) \(\frac{1}{2} \left(1+\sqrt{5}\right)\)
\(\mathbf{5}\) \(2.\) \(2\)
\(\mathbf{6}\) \(3.23607\) \(1+\sqrt{5}\)
\(\mathcal{D}_{FP}^2\) \(21.7082\) \(6+\frac{3}{2} \left(1+\sqrt{5}\right)^2\)

Characters

The symbolic character table is the following

\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} \\ \hline 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 2 & 1+\sqrt{5} \\ 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 2 & 1-\sqrt{5} \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) \\ 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 0 & 0 \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 0 & 0 \\ \hline \end{array}\]

The numeric character table is the following

\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.618 & 1.618 & 2.000 & 3.236 \\ 1.000 & 1.000 & 1.618 & 1.618 & -1.000 & -1.618 \\ 1.000 & 1.000 & -0.6180 & -0.6180 & 2.000 & -1.236 \\ 1.000 & 1.000 & -0.6180 & -0.6180 & -1.000 & 0.6180 \\ 1.000 & -1.000 & 1.618 & -1.618 & 0 & 0 \\ 1.000 & -1.000 & -0.6180 & 0.6180 & 0 & 0 \\ \hline \end{array}\]

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

Data

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