\(\text{SU(2})_5:\ \text{FR}^{6,0}_{6}\)
Fusion Rules
\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{5} & \mathbf{2}+\mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{6} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{6} & \mathbf{1}+\mathbf{5} & \mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{5} \\ \mathbf{5} & \mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{6} & \mathbf{1}+\mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{4}+\mathbf{6} \\ \mathbf{6} & \mathbf{5} & \mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{4}+\mathbf{6} & \mathbf{1}+\mathbf{3}+\mathbf{5} \\ \hline \end{array}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{3},\mathbf{5}\}\) | \(\text{PSU}(2)_5:\ \text{FR}^{3,0}_{3}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.80194\) | \(\sin\frac{5\pi}{7}/\sin\frac{\pi}{7}\) |
| \(\mathbf{4}\) | \(1.80194\) | \(\sin\frac{5\pi}{7}/\sin\frac{\pi}{7}\) |
| \(\mathbf{5}\) | \(2.24698\) | \(\sin\frac{3\pi}{7}/\sin\frac{\pi}{7}\) |
| \(\mathbf{6}\) | \(2.24698\) | \(\sin\frac{3\pi}{7}/\sin\frac{\pi}{7}\) |
| \(\mathcal{D}_{FP}^2\) | \(18.5918\) | \(\frac{7}{2\sin(\pi/7)^2}\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} \\ \hline 1 & 1 & a_3 & a_3 & b_3 & b_3 \\ 1 & 1 & a_2 & a_2 & b_1 & b_1 \\ 1 & 1 & a_1 & a_1 & b_2 & b_2 \\ 1 & -1 & a_3 & -a_3 & -b_3 & b_3 \\ 1 & -1 & a_2 & -a_2 & -b_1 & b_1 \\ 1 & -1 & a_1 & -a_1 & -b_2 & b_2 \\ \hline \end{array}\]where $a_i$ and $b_i$ are the $i’$th roots of the polynomials $x^3 - x^2 -2x + 1$ and $x^3 -2x^2 -x + 1$.
The numeric character table is the following
\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} \\ \hline 1.000 & 1.000 & 1.802 & 1.802 & 2.247 & 2.247 \\ 1.000 & 1.000 & 0.4450 & 0.4450 & -0.8019 & -0.8019 \\ 1.000 & 1.000 & -1.247 & -1.247 & 0.5550 & 0.5550 \\ 1.000 & -1.000 & 1.802 & -1.802 & -2.247 & 2.247 \\ 1.000 & -1.000 & 0.4450 & -0.4450 & 0.8019 & -0.8019 \\ 1.000 & -1.000 & -1.247 & 1.247 & -0.5550 & 0.5550 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{2\sin(\pi/7)}{7}\left( \begin{array}{ccccc} 1 & 1 & d_3 & d_3 & d_5 & d_5 \\ 1 & -1 & d_3 & -d_3 & d_5 & -d_5 \\ d_3 & d_3 & -d_5 & -d_5 & 1 & 1 \\ d_3 & -d_3 & -d_5 & d_5 & 1 & -1 \\ d_5 & d_5 & 1 & 1 & - d_3 & -d_3 \\ d_5 & -d_5 & 1 & -1 & -d_3 & d_3 \end{array}\right)\) | \(\begin{array}{l}\left(0,-\frac{1}{4},-\frac{1}{7},-\frac{11}{28},\frac{2}{7},\frac{1}{28}\right) \\\left(0,-\frac{1}{4},\frac{1}{7},-\frac{3}{28},-\frac{2}{7},\frac{13}{28}\right) \\\left(0,\frac{1}{4},-\frac{1}{7},\frac{3}{28},\frac{2}{7},-\frac{13}{28}\right) \\\left(0,\frac{1}{4},\frac{1}{7},\frac{11}{28},-\frac{2}{7},-\frac{1}{28}\right)\end{array}\) |
where $d_3$ and $d_5$ are the quantum dimensions of resp the $3’$rd and $5’th$ particles.
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{3}, \mathbf{5}\), form the adjoint subring \(\text{PSU}(2)_5:\ \text{FR}^{3,0}_{3}\) .
The upper central series is the following: \(\text{SU(2})_5 \underset{ \mathbf{1}, \mathbf{3}, \mathbf{5} }{\supset} \text{PSU}(2)_5\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{2}'\), 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
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