\(\text{Rep(}D_5\text{)$\times $}\mathbb{Z}_2:\ \text{FR}^{8,0}_{3}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{5} & \mathbf{8} & \mathbf{8} & \mathbf{5} & \mathbf{1}+\mathbf{4}+\mathbf{7} & \mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{6} \\ \mathbf{6} & \mathbf{7} & \mathbf{7} & \mathbf{6} & \mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5} & \mathbf{5}+\mathbf{6} \\ \mathbf{7} & \mathbf{6} & \mathbf{6} & \mathbf{7} & \mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{5} & \mathbf{1}+\mathbf{4}+\mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{5} & \mathbf{5} & \mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{7} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}), (\mathbf{5} \ \mathbf{6}) (\mathbf{7} \ \mathbf{8})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{3}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4},\mathbf{7},\mathbf{8}\}\) | \(\left.\text{Rep(}D_5\right):\ \text{FR}^{4,0}_{3}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(2.\) | \(2\) |
| \(\mathbf{6}\) | \(2.\) | \(2\) |
| \(\mathbf{7}\) | \(2.\) | \(2\) |
| \(\mathbf{8}\) | \(2.\) | \(2\) |
| \(\mathcal{D}_{FP}^2\) | \(20.\) | \(20\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 \\ 1 & 1 & 1 & 1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) \\ 1 & 1 & 1 & 1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & -1 & 1 & -1 & -2 & -2 & 2 & 2 \\ 1 & -1 & 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) \\ 1 & -1 & 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & 0.6180 & -1.618 & -1.618 & 0.6180 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.618 & 0.6180 & 0.6180 & -1.618 \\ 1.000 & -1.000 & 1.000 & -1.000 & -2.000 & -2.000 & 2.000 & 2.000 \\ 1.000 & -1.000 & 1.000 & -1.000 & -0.6180 & 1.618 & -1.618 & 0.6180 \\ 1.000 & -1.000 & 1.000 & -1.000 & 1.618 & -0.6180 & 0.6180 & -1.618 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 & 0 & 0 \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{4}, \mathbf{7}, \mathbf{8}\), form the adjoint subring \(\left.\text{Rep(}D_5\right):\ \text{FR}^{4,0}_{3}\) .
The upper central series is the following: \(\text{Rep(}D_5\text{)$\times $}\mathbb{Z}_2 \underset{ \mathbf{1}, \mathbf{4}, \mathbf{7}, \mathbf{8} }{\supset} \left.\text{Rep(}D_5\right)\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{2}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{1}', \text{deg}(\mathbf{8}) = \mathbf{1}'\), 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: