\(\text{PSU}(2)_5:\ \text{FR}^{3,0}_{3}\)
Fusion Rules
\[\begin{array}{|lll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \mathbf{2} & \mathbf{1}+\mathbf{3} & \mathbf{2}+\mathbf{3} \\ \mathbf{3} & \mathbf{2}+\mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} \\ \hline \end{array}\]Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(\sin\frac{\pi}{7}/\sin\frac{\pi}{7}\) |
| \(\mathbf{2}\) | \(1.80194\) | \(\sin\frac{5\pi}{7}/\sin\frac{\pi}{7}\) |
| \(\mathbf{3}\) | \(2.24698\) | \(\sin\frac{3\pi}{7}/\sin\frac{\pi}{7}\) |
| \(\mathcal{D}_{FP}^2\) | \(9.2959\) | \(\frac{7}{4\sin(\pi/7)}\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline 1 & r_3 & r'_3 \\ 1 & r_2 & r'_2 \\ 1 & r_1 & r'_1 \\ \hline \end{array}\]where \(r_i\) is the \(i\)‘th root of \(x^3-x^2-2 x+1\) and \(r'_i\) the \(i\)‘th root of \(x^2-2x^2-x+1\)
The numeric character table is the following
\[\begin{array}{|rrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline 1.000 & 1.802 & 2.247 \\ 1.000 & 0.4450 & -0.8019 \\ 1.000 & -1.247 & 0.5550 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{2\sin(\pi/7)}{\sqrt{7}}\left(\begin{array}{ccc} 1 & D_2 & D_3 \\ D_2 & -D_3 & 1 \\ D_3 & 1 & -D_2 \end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{7},-\frac{2}{7}\right) \\\left(0,-\frac{1}{7},\frac{2}{7}\right)\end{array}\) |
where $D_i$ are the $i$’th Frobenius-Perron dimensions.
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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