\(\text{PSU(2})_7:\ \text{FR}^{4,0}_{6}\)
Fusion Rules
\[\begin{array}{|llll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \mathbf{2} & \mathbf{1}+\mathbf{3} & \mathbf{2}+\mathbf{4} & \mathbf{3}+\mathbf{4} \\ \mathbf{3} & \mathbf{2}+\mathbf{4} & \mathbf{1}+\mathbf{3}+\mathbf{4} & \mathbf{2}+\mathbf{3}+\mathbf{4} \\ \mathbf{4} & \mathbf{3}+\mathbf{4} & \mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} \\ \hline \end{array}\]Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.87939\) | $ \sin( 7 \pi/9)/\sin(\pi/9) $ |
| \(\mathbf{3}\) | \(2.53209\) | $ \sin( 3 \pi/9)/\sin(\pi/9) $ |
| \(\mathbf{4}\) | \(2.87939\) | $ \sin( 5 \pi/9)/\sin(\pi/9) $ |
| \(\mathcal{D}_{FP}^2\) | \(19.2344\) | $ \frac{9}{4\sin(\pi/9)^2} $ |
Characters
The symbolic character table is the following
\[\begin{array}{|cccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \hline 1 & a_3 & b_3 & c_3 \\ 1 & 1 & 0 & -1 \\ 1 & a_2 & b_1 & c_2 \\ 1 & a_1 & b_2 & c_1 \\ \hline \end{array}\]where \(a_i := \text{Root}[ -1 -3x + x^3, i ]\), \(b_i := \text{Root}[ 3 -3x^2 + x^3, i ]\), and \(c_i := \text{Root}[ 1 -3x^2 + x^3, i ]\). In particular \(a_3 = \sin( 7 \pi/9)/\sin(\pi/9)\), \(b_3 = \sin( 3 \pi/9)/\sin(\pi/9)\), and \(c_3 = \sin( 5 \pi/9)/\sin(\pi/9)\).
The numeric character table is the following
\[\begin{array}{|rrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \hline 1.000 & 1.879 & 2.532 & 2.879 \\ 1.000 & 1.000 & 0 & -1.000 \\ 1.000 & -0.3473 & -0.8794 & 0.6527 \\ 1.000 & -1.532 & 1.347 & -0.5321 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{2\sin\left(\frac{\pi}{9}\right)}{3}\left(\begin{array}{cccc} 1 & D_2 & D_3 & D_4\\ D_2 & -D_4 & D_3 & -1 \\ D_3 & D_3 & 0 & -D_3 \\ D_4 & -1 & -D_3 & D_2 \end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{3},\frac{2}{9},-\frac{1}{3}\right) \\\left(0,-\frac{1}{3},-\frac{2}{9},\frac{1}{3}\right)\end{array}\) |
where $D_i$ stands for 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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