\(\text{PSU(2})_9:\ \text{FR}^{5,0}_{10}\)

Fusion Rules

\[\begin{array}{|lllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \mathbf{2} & \mathbf{1}+\mathbf{3} & \mathbf{2}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{5} \\ \mathbf{3} & \mathbf{2}+\mathbf{4} & \mathbf{1}+\mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{5} \\ \mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\ \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\ \hline \end{array}\]

Quantum Dimensions

Particle Numeric Symbolic
\(\mathbf{1}\) \(1.\) \(\sin(\pi/11)/\sin(\pi/11)\)
\(\mathbf{2}\) \(1.91899\) \(\sin(9\pi/11)/\sin(\pi/11)\)
\(\mathbf{3}\) \(2.68251\) \(\sin(3\pi/11)/\sin(\pi/11)\)
\(\mathbf{4}\) \(3.22871\) \(\sin(7\pi/11)/\sin(\pi/11)\)
\(\mathbf{5}\) \(3.51334\) \(\sin(5\pi/11)/\sin(\pi/11)\)
\(\mathcal{D}_{FP}^2\) \(34.6464\) \(\frac{11}{2\sin(\pi/11)}\)

Characters

The symbolic character table is the following

\[\begin{array}{|ccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline 1 & a_5 & b_5 & c_5 & d_5 \\ 1 & a_4 & b_3 & c_3 & d_1 \\ 1 & a_3 & b_1 & c_2 & d_4 \\ 1 & a_2 & b_2 & c_4 & d_2 \\ 1 & a_1 & b_4 & c_1 & d_3 \\ \hline \end{array}\]

where the \(a_i\), \(b_i\), \(c_i\), \(d_i\) are respectively the $i’$th roots of the polynomials

The numeric character table is the following

\[\begin{array}{|rrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline 1.000 & 1.919 & 2.683 & 3.229 & 3.513 \\ 1.000 & 1.310 & 0.7154 & -0.3728 & -1.204 \\ 1.000 & 0.2846 & -0.9190 & -0.5462 & 0.7635 \\ 1.000 & -0.8308 & -0.3097 & 1.088 & -0.5944 \\ 1.000 & -1.683 & 1.831 & -1.398 & 0.5211 \\ \hline \end{array}\]

Modular Data

The matching \(S\)-matrices and twist factors are the following

\(S\)-matrix Twist factors
\(\frac{2\sin(\pi/11)}{\sqrt{11}}\left(\begin{array}{ccccc} 1 & D_2 & D_3 & D_4 & D_5 \\ D_2 & -D_4 & D_5 & -D_3 & 1 \\ D_3 & D_5 & D_2 & -1 & -D_4 \\ D_4 & -D_3 & -1 & D_5 & -d2 \\ D_5 & 1 & -D_4 & -D_2 & D_3 \end{array}\right)\) \(\begin{array}{l}\left(0,\frac{2}{11},-\frac{2}{11},-\frac{1}{11},\frac{5}{11}\right) \\\left(0,-\frac{2}{11},\frac{2}{11},\frac{1}{11},-\frac{5}{11}\right)\end{array}\)

where $D_i$ is the $i’$th quantum dimension.

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

Download links for numeric data: