\(\text{PSU}(2)_5:\ \text{FR}^{3,0}_{3}\)

Fusion Rules

\[\begin{array}{|lll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \mathbf{2} & \mathbf{1}+\mathbf{3} & \mathbf{2}+\mathbf{3} \\ \mathbf{3} & \mathbf{2}+\mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} \\ \hline \end{array}\]

Quantum 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

\[\begin{array}{|rrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline 1.000 & 1.802 & 2.247 \\ 1.000 & 0.4450 & -0.8019 \\ 1.000 & -1.247 & 0.5550 \\ \hline \end{array}\]

Modular Data

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 quantum 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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