\(\text{SU(2})_4:\ \text{FR}^{5,0}_{3}\)
Fusion Rules
\[\begin{array}{|lllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{5} & \mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{5} & \mathbf{1}+\mathbf{5} & \mathbf{3}+\mathbf{4} \\ \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{5} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{3} \ \mathbf{4})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{5}\}\) | \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.73205\) | \(\sqrt{3}\) |
| \(\mathbf{4}\) | \(1.73205\) | \(\sqrt{3}\) |
| \(\mathbf{5}\) | \(2.\) | \(2\) |
| \(\mathcal{D}_{FP}^2\) | \(12.\) | \(12\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & 0 & 0 & -1 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ 1 & -1 & 1 & -1 & 0 \\ 1 & -1 & -1 & 1 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline 1.000 & 1.000 & 1.732 & 1.732 & 2.000 \\ 1.000 & 1.000 & 0 & 0 & -1.000 \\ 1.000 & 1.000 & -1.732 & -1.732 & 2.000 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{2 \sqrt{3}}\left(\begin{array}{ccccc} 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ \sqrt{3} & -\sqrt{3} & \sqrt{3} & -\sqrt{3} & 0 \\ \sqrt{3} & -\sqrt{3} & -\sqrt{3} & \sqrt{3} & 0 \\ 2 & 2 & 0 & 0 & -2 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,\frac{3}{8},-\frac{1}{8},-\frac{1}{3}\right) \\\left(0,0,-\frac{1}{8},\frac{3}{8},-\frac{1}{3}\right) \\\left(0,0,\frac{1}{8},-\frac{3}{8},\frac{1}{3}\right) \\\left(0,0,-\frac{3}{8},\frac{1}{8},\frac{1}{3}\right)\end{array}\) |
| \(\frac{1}{2 \sqrt{3}}\left(\begin{array}{ccccc} 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ \sqrt{3} & -\sqrt{3} & -\sqrt{3} & \sqrt{3} & 0 \\ \sqrt{3} & -\sqrt{3} & \sqrt{3} & -\sqrt{3} & 0 \\ 2 & 2 & 0 & 0 & -2 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,-\frac{1}{8},\frac{3}{8},\frac{1}{3}\right) \\\left(0,0,\frac{3}{8},-\frac{1}{8},\frac{1}{3}\right) \\\left(0,0,-\frac{3}{8},\frac{1}{8},-\frac{1}{3}\right) \\\left(0,0,\frac{1}{8},-\frac{3}{8},-\frac{1}{3}\right)\end{array}\) |
Adjoint Subring
Elements \(\mathbf{1}, \mathbf{2}, \mathbf{5}\), form the adjoint subring \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) .
The upper central series is the following: \(\text{SU(2})_4 \underset{ \mathbf{1}, \mathbf{2}, \mathbf{5} }{\supset} \left.\text{Rep(}D_3\right)\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}'\), 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
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