\(\text{SU}(2)_7:\ \text{FR}^{8,0}_{15}\)

Fusion Rules

\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{5} & \mathbf{2}+\mathbf{6} & \mathbf{3}+\mathbf{7} & \mathbf{4}+\mathbf{8} & \mathbf{5}+\mathbf{7} & \mathbf{6}+\mathbf{8} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{6} & \mathbf{1}+\mathbf{5} & \mathbf{4}+\mathbf{8} & \mathbf{3}+\mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{7} \\ \mathbf{5} & \mathbf{6} & \mathbf{3}+\mathbf{7} & \mathbf{4}+\mathbf{8} & \mathbf{1}+\mathbf{5}+\mathbf{7} & \mathbf{2}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{8} \\ \mathbf{6} & \mathbf{5} & \mathbf{4}+\mathbf{8} & \mathbf{3}+\mathbf{7} & \mathbf{2}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{7} \\ \mathbf{7} & \mathbf{8} & \mathbf{5}+\mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{6}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{7} \\ \hline \end{array}\]

The following particles form non-trivial sub fusion rings

Particles	SubRing
\(\{\mathbf{1},\mathbf{2}\}\)	\(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\)
\(\{\mathbf{1},\mathbf{3},\mathbf{5},\mathbf{7}\}\)	\(\text{PSU(2})_7:\ \text{FR}^{4,0}_{6}\)

Quantum Dimensions

Particle	Numeric	Symbolic
\(\mathbf{1}\)	\(1.\)	\(1\)
\(\mathbf{2}\)	\(1.\)	\(1\)
\(\mathbf{3}\)	\(1.87939\)	\(\text{Root}\left[x^3-3 x-1,3\right]\)
\(\mathbf{4}\)	\(1.87939\)	\(\text{Root}\left[x^3-3 x-1,3\right]\)
\(\mathbf{5}\)	\(2.53209\)	\(\text{Root}\left[x^3-3 x^2+3,3\right]\)
\(\mathbf{6}\)	\(2.53209\)	\(\text{Root}\left[x^3-3 x^2+3,3\right]\)
\(\mathbf{7}\)	\(2.87939\)	\(\text{Root}\left[x^3-3 x^2+1,3\right]\)
\(\mathbf{8}\)	\(2.87939\)	\(\text{Root}\left[x^3-3 x^2+1,3\right]\)
\(\mathcal{D}_{FP}^2\)	\(38.4688\)	\(2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2\)

Characters

The symbolic character table is the following

\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \hline 1 & 1 & \text{Root}\left[x^3-3 x-1,3\right] & \text{Root}\left[x^3-3 x-1,3\right] & \text{Root}\left[x^3-3 x^2+3,3\right] & \text{Root}\left[x^3-3 x^2+3,3\right] & \text{Root}\left[x^3-3 x^2+1,3\right] & \text{Root}\left[x^3-3 x^2+1,3\right] \\ 1 & 1 & 1 & 1 & 0 & 0 & -1 & -1 \\ 1 & 1 & \text{Root}\left[x^3-3 x-1,2\right] & \text{Root}\left[x^3-3 x-1,2\right] & \text{Root}\left[x^3-3 x^2+3,1\right] & \text{Root}\left[x^3-3 x^2+3,1\right] & \text{Root}\left[x^3-3 x^2+1,2\right] & \text{Root}\left[x^3-3 x^2+1,2\right] \\ 1 & 1 & \text{Root}\left[x^3-3 x-1,1\right] & \text{Root}\left[x^3-3 x-1,1\right] & \text{Root}\left[x^3-3 x^2+3,2\right] & \text{Root}\left[x^3-3 x^2+3,2\right] & \text{Root}\left[x^3-3 x^2+1,1\right] & \text{Root}\left[x^3-3 x^2+1,1\right] \\ 1 & -1 & \text{Root}\left[x^3-3 x-1,3\right] & \text{Root}\left[x^3-3 x+1,1\right] & \text{Root}\left[x^3-3 x^2+3,3\right] & \text{Root}\left[x^3+3 x^2-3,1\right] & \text{Root}\left[x^3+3 x^2-1,1\right] & \text{Root}\left[x^3-3 x^2+1,3\right] \\ 1 & -1 & 1 & -1 & 0 & 0 & 1 & -1 \\ 1 & -1 & \text{Root}\left[x^3-3 x-1,1\right] & \text{Root}\left[x^3-3 x+1,3\right] & \text{Root}\left[x^3-3 x^2+3,2\right] & \text{Root}\left[x^3+3 x^2-3,2\right] & \text{Root}\left[x^3+3 x^2-1,3\right] & \text{Root}\left[x^3-3 x^2+1,1\right] \\ 1 & -1 & \text{Root}\left[x^3-3 x-1,2\right] & \text{Root}\left[x^3-3 x+1,2\right] & \text{Root}\left[x^3-3 x^2+3,1\right] & \text{Root}\left[x^3+3 x^2-3,3\right] & \text{Root}\left[x^3+3 x^2-1,2\right] & \text{Root}\left[x^3-3 x^2+1,2\right] \\ \hline \end{array}\]

The numeric character table is the following

\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \hline 1.000 & 1.000 & 1.879 & 1.879 & 2.532 & 2.532 & 2.879 & 2.879 \\ 1.000 & 1.000 & 1.000 & 1.000 & 0 & 0 & -1.000 & -1.000 \\ 1.000 & 1.000 & -0.3473 & -0.3473 & -0.8794 & -0.8794 & 0.6527 & 0.6527 \\ 1.000 & 1.000 & -1.532 & -1.532 & 1.347 & 1.347 & -0.5321 & -0.5321 \\ 1.000 & -1.000 & 1.879 & -1.879 & 2.532 & -2.532 & -2.879 & 2.879 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 & 0 & 1.000 & -1.000 \\ 1.000 & -1.000 & -1.532 & 1.532 & 1.347 & -1.347 & 0.5321 & -0.5321 \\ 1.000 & -1.000 & -0.3473 & 0.3473 & -0.8794 & 0.8794 & -0.6527 & 0.6527 \\ \hline \end{array}\]

Modular Data

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

\(S\)-matrix	Twist factors
\(\frac{1}{\sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}\left(\begin{array}{cccccccc} \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & 0 & 0 & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} \\ \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & 0 & 0 & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\\end{array}\right)\)	\(\begin{array}{l}\left(0,\frac{1}{4},-\frac{1}{3},-\frac{1}{12},-\frac{2}{9},\frac{1}{36},\frac{1}{3},-\frac{5}{12}\right) \\\left(0,\frac{1}{4},\frac{1}{3},-\frac{5}{12},\frac{2}{9},\frac{17}{36},-\frac{1}{3},-\frac{1}{12}\right) \\\left(0,-\frac{1}{4},-\frac{1}{3},\frac{5}{12},-\frac{2}{9},-\frac{17}{36},\frac{1}{3},\frac{1}{12}\right) \\\left(0,-\frac{1}{4},\frac{1}{3},\frac{1}{12},\frac{2}{9},-\frac{1}{36},-\frac{1}{3},\frac{5}{12}\right)\end{array}\)

\(S\)-matrix

Twist factors

\(\frac{1}{\sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}\left(\begin{array}{cccccccc} \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & 0 & 0 & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} \\ \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & 0 & 0 & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\ \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,3\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & -\frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \frac{1}{\sqrt{\frac{6}{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2}}} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} & \text{Root}\left[1944 x^6-648 x^4+54 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-3 x-1,3\right]^2+2 \text{Root}\left[x^3-3 x^2+3,3\right]^2+2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2} \\\end{array}\right)\)

\(\begin{array}{l}\left(0,\frac{1}{4},-\frac{1}{3},-\frac{1}{12},-\frac{2}{9},\frac{1}{36},\frac{1}{3},-\frac{5}{12}\right) \\\left(0,\frac{1}{4},\frac{1}{3},-\frac{5}{12},\frac{2}{9},\frac{17}{36},-\frac{1}{3},-\frac{1}{12}\right) \\\left(0,-\frac{1}{4},-\frac{1}{3},\frac{5}{12},-\frac{2}{9},-\frac{17}{36},\frac{1}{3},\frac{1}{12}\right) \\\left(0,-\frac{1}{4},\frac{1}{3},\frac{1}{12},\frac{2}{9},-\frac{1}{36},-\frac{1}{3},\frac{5}{12}\right)\end{array}\)

Adjoint Subring

Particles \(\mathbf{1}, \mathbf{3}, \mathbf{5}, \mathbf{7}\), form the adjoint subring \(\text{PSU(2})_7:\ \text{FR}^{4,0}_{6}\) .

The upper central series is the following: \(\text{SU}(2)_7 \underset{ \mathbf{1}, \mathbf{3}, \mathbf{5}, \mathbf{7} }{\supset} \text{PSU(2})_7\)

Universal grading

Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{1}', \text{deg}(\mathbf{8}) = \mathbf{2}'\), where the degrees form the group \(\mathbb{Z}_2\) with multiplication table:

\[\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}\]

Categorifications

Data

Download links for numeric data:

AnyonWiki