\(\text{Fib} \times \text{PSU}(2)_5 :\ \text{FR}^{6,0}_{14}\)
Fusion Rules
\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{1}+\mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{6} \\ \mathbf{3} & \mathbf{5} & \mathbf{1}+\mathbf{4} & \mathbf{3}+\mathbf{4} & \mathbf{2}+\mathbf{6} & \mathbf{5}+\mathbf{6} \\ \mathbf{4} & \mathbf{6} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{3}+\mathbf{4} & \mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{5}+\mathbf{6} \\ \mathbf{5} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{6} & \mathbf{4}+\mathbf{6} & \mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \hline \end{array}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{3},\mathbf{4}\}\) | \(\text{PSU}(2)_5:\ \text{FR}^{3,0}_{3}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.61803\) | \(\phi\) |
| \(\mathbf{3}\) | \(1.80194\) | \(\frac{\sin(5\pi/7)}{\sin(\pi/7)}\) |
| \(\mathbf{4}\) | \(2.24698\) | \(\frac{\sin(3\pi/7)}{\sin(\pi/7)}\) |
| \(\mathbf{5}\) | \(2.9156\) | \(\phi \frac{\sin(5\pi/7)}{\sin(\pi/7)}\) |
| \(\mathbf{6}\) | \(3.63569\) | \(\phi\frac{\sin(3\pi/7)}{\sin(\pi/7)}\) |
| \(\mathcal{D}_{FP}^2\) | \(33.6329\) | \(\phi^2 \frac{7}{4 \left(\sin\frac{\pi }{7}\right)^2}\) |
Here $\phi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio.
Characters
The symbolic character table is the following
\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1 & \phi & a_3 & b_3 & a_3 \phi & b_3 \phi \\ 1 & -\phi^{-1} & a_3 & b_3 & - a_3 \phi^{-1} & -b_3 \phi^{-1} \\ 1 & \phi & a_2 & b_1 & a_2 \phi & b_1 \phi \\ 1 & \phi & a_1 & b_2 & a_1 \phi & b_2 \phi \\ 1 & -\phi^{-1} & a_2 & b_1 & - a_2 \phi^{-1} & -b_1 \phi^{-1} \\ 1 & -\phi^{-1} & a_1 & b_2 & -a_1 \phi^{-1} & -b_2 \phi^{-1} \\ \hline \end{array}\]Where $a_i$ is the $i$’th root of the polynomial $x^3-x^2-2 x+1$ and $b_i$ is the $i$’th root of the polynomial $x^3-2 x^2-x+1$.
The numeric character table is the following
\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1.000 & 1.618 & 1.802 & 2.247 & 2.916 & 3.636 \\ 1.000 & -0.6180 & 1.802 & 2.247 & -1.114 & -1.389 \\ 1.000 & 1.618 & 0.4450 & -0.8019 & 0.7201 & -1.298 \\ 1.000 & 1.618 & -1.247 & 0.5550 & -2.018 & 0.8979 \\ 1.000 & -0.6180 & 0.4450 & -0.8019 & -0.2751 & 0.4956 \\ 1.000 & -0.6180 & -1.247 & 0.5550 & 0.7707 & -0.3430 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{ 2 \sin \left(\frac{\pi }{7}\right) }{\phi\sqrt{7}} \left(\begin{array}{cccccc} 1 & D_2 & D_3 & D_4 & D_5 & D_6 \\ D_2 & -1 & D_5 & D_6 & - D_3 & -D_4 \\ D_3 & D_5 & -D_4 & 1 & -D_6 & D_2\\ D_4 & D_6 & 1 & - D_3 & D_2 & -D_5 \\ D_5 & -D_3 & -D_6 & D_2 & D_4 & -1 \\ D_6 & -D_4 & D_2 & -D_5 & -1 & -D_3 \end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{2}{5},-\frac{1}{7},\frac{2}{7},\frac{9}{35},-\frac{11}{35}\right) \\\left(0,-\frac{2}{5},-\frac{1}{7},\frac{2}{7},\frac{16}{35},-\frac{4}{35}\right) \\\left(0,\frac{2}{5},\frac{1}{7},-\frac{2}{7},-\frac{16}{35},\frac{4}{35}\right) \\\left(0,-\frac{2}{5},\frac{1}{7},-\frac{2}{7},-\frac{9}{35},\frac{11}{35}\right)\end{array}\) |
Where $D_i$ is $i’$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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