\(\text{PSU}(2)_{11}:\ \text{FR}^{6,0}_{18}\)
Fusion Rules
\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{1}+\mathbf{3} & \mathbf{2}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{6} & \mathbf{5}+\mathbf{6} \\ \mathbf{3} & \mathbf{2}+\mathbf{4} & \mathbf{1}+\mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{4}+\mathbf{6} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{5} & \mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \hline \end{array}\]Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(\frac{\sin{\pi/13}}{\sin{\pi/13}}\) |
| \(\mathbf{2}\) | \(1.94188\) | \(\frac{\sin{11\pi/13}}{\sin{\pi/13}}\) |
| \(\mathbf{3}\) | \(2.77091\) | \(\frac{\sin{3\pi/13}}{\sin{\pi/13}}\) |
| \(\mathbf{4}\) | \(3.43891\) | \(\frac{\sin{9\pi/13}}{\sin{\pi/13}}\) |
| \(\mathbf{5}\) | \(3.90704\) | \(\frac{\sin{5\pi/13}}{\sin{\pi/13}}\) |
| \(\mathbf{6}\) | \(4.14811\) | \(\frac{\sin{7\pi/13}}{\sin{\pi/13}}\) |
| \(\mathcal{D}_{FP}^2\) | \(56.7468\) | \(\frac{13}{4\left(\sin(\pi/13)\right)^2}\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1 & a_6 & b_6 & c_6 & D_6 & e_6 \\ 1 & a_5 & b_4 & c_3 & D_2 & e_1 \\ 1 & a_4 & b_2 & c_2 & D_3 & e_5 \\ 1 & a_3 & b_1 & c_4 & D_4 & e_2 \\ 1 & a_2 & b_3 & c_5 & D_1 & e_4 \\ 1 & a_1 & b_5 & c_1 & D_5 & e_3 \\ \hline \end{array}\]where $a_i,b_i,c_i,D_i,e_i$ are resp. the $i$’th roots of the following polynomials
- $x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1$
- $x^6-5 x^5+5 x^4+6 x^3-7 x^2-2 x+1$
- $x^6-5 x^5+5 x^4+6 x^3-7 x^2-2 x+1$
- $x^6-2 x^5-7 x^4+6 x^3+5 x^2-5 x+1$
- $x^6-4 x^5-2 x^4+9 x^3+2 x^2-4 x-1$
- $x^6-3 x^5-6 x^4+4 x^3+5 x^2-x-1$
The numeric character table is the following
\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1.000 & 1.942 & 2.771 & 3.439 & 3.907 & 4.148 \\ 1.000 & 1.497 & 1.241 & 0.3609 & -0.7008 & -1.410 \\ 1.000 & 0.7092 & -0.4970 & -1.062 & -0.2559 & 0.8802 \\ 1.000 & -0.2411 & -0.9419 & 0.4681 & 0.8290 & -0.6680 \\ 1.000 & -1.136 & 0.2908 & 0.8058 & -1.206 & 0.5647 \\ 1.000 & -1.771 & 2.136 & -2.012 & 1.427 & -0.5150 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{2 \sin\frac{\pi}{13}}{\sqrt{13}}\left( \begin{array}{cccccc} 1 & D_2 & D_3 & D_4 & D_5 & D_6 \\ D_2 & - D_4 & D_6 & -D_5 & D_3 & -1 \\ D_3 & D_6 & D_4 & 1 & -D_2 & - D_5 \\ D_4 & -D_5 & 1 & D_3 & -D_6 & D_2 \\ D_5 & D_3 & -D_2 & -D_6 & -1 & D_4 \\ D_6 & -1 & -D_5 & D_1 & D_4 & -D_3 \end{array} \right)\) | \(\begin{array}{l}\left(0,\frac{4}{13},\frac{2}{13},-\frac{6}{13},\frac{6}{13},-\frac{1}{13}\right) \\\left(0,-\frac{4}{13},-\frac{2}{13},\frac{6}{13},-\frac{6}{13},\frac{1}{13}\right)\end{array}\) |
where $D_i$ is the Frobenius-Perron dimension of the $i$’th particle.
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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