\(\text{PSU(2})_9:\ \text{FR}^{5,0}_{10}\)
Fusion Rules
\[\begin{array}{|lllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \mathbf{2} & \mathbf{1}+\mathbf{3} & \mathbf{2}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{5} \\ \mathbf{3} & \mathbf{2}+\mathbf{4} & \mathbf{1}+\mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{5} \\ \mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\ \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\ \hline \end{array}\]Frobenius-Perron 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
- \(x^5-x^4-4 x^3+3 x^2+3 x-1\),
- \(x^5-4 x^4+2 x^3+5 x^2-2 x-1\),
- \(x^5-2 x^4-5 x^3+2 x^2+4 x+1\),
- \(x^5-3 x^4-3 x^3+4 x^2+x-1\).
The numeric character table is the following
\[\begin{array}{|rrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline 1.000 & 1.919 & 2.683 & 3.229 & 3.513 \\ 1.000 & 1.310 & 0.7154 & -0.3728 & -1.204 \\ 1.000 & 0.2846 & -0.9190 & -0.5462 & 0.7635 \\ 1.000 & -0.8308 & -0.3097 & 1.088 & -0.5944 \\ 1.000 & -1.683 & 1.831 & -1.398 & 0.5211 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
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 Frobenius-Perron 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
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