\(\text{SU}(2)_6:\ \text{FR}^{7,0}_{7}\)

Fusion Rules

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

The fusion rules are invariant under the group generated by the following permutations:

\[\{(\mathbf{3} \ \mathbf{4})\}\]

The following particles form non-trivial sub fusion rings

Particles	SubRing
\(\{\mathbf{1},\mathbf{2}\}\)	\(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\)
\(\{\mathbf{1},\mathbf{2},\mathbf{5},\mathbf{6}\}\)	\(\text{PSU(2})_6:\ \text{FR}^{4,0}_{4}\)

Quantum Dimensions

Particle	Numeric	Symbolic
\(\mathbf{1}\)	\(1.\)	\(1\)
\(\mathbf{2}\)	\(1.\)	\(1\)
\(\mathbf{3}\)	\(1.84776\)	\(\sqrt{2+\sqrt{2}}\)
\(\mathbf{4}\)	\(1.84776\)	\(\sqrt{2+\sqrt{2}}\)
\(\mathbf{5}\)	\(2.41421\)	\(1+\sqrt{2}\)
\(\mathbf{6}\)	\(2.41421\)	\(1+\sqrt{2}\)
\(\mathbf{7}\)	\(2.61313\)	\(\sqrt{2 \left(2+\sqrt{2}\right)}\)
\(\mathcal{D}_{FP}^2\)	\(27.3137\)	\(6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)\)

Characters

The symbolic character table is the following

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

The numeric character table is the following

\[\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{7} \\ \hline 1.000 & 1.000 & 1.848 & 1.848 & 2.414 & 2.414 & 2.613 \\ 1.000 & 1.000 & 0.7654 & 0.7654 & -0.4142 & -0.4142 & -1.082 \\ 1.000 & 1.000 & -0.7654 & -0.7654 & -0.4142 & -0.4142 & 1.082 \\ 1.000 & 1.000 & -1.848 & -1.848 & 2.414 & 2.414 & -2.613 \\ 1.000 & -1.000 & 1.414 & -1.414 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & -1.414 & 1.414 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & 0 & 0 & -1.000 & 1.000 & 0 \\ \hline \end{array}\]

Modular Data

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

\(S\)-matrix	Twist factors
\(\frac{1}{\sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}\left(\begin{array}{ccccccc} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & 0 \\ \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & 0 \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & 0 & 0 & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & 0 \\\end{array}\right)\)	\(\begin{array}{l}\left(0,\frac{1}{2},\frac{13}{32},\frac{13}{32},\frac{1}{4},-\frac{1}{4},\frac{1}{32}\right) \\\left(0,\frac{1}{2},\frac{3}{32},\frac{3}{32},-\frac{1}{4},\frac{1}{4},\frac{15}{32}\right) \\\left(0,\frac{1}{2},-\frac{3}{32},-\frac{3}{32},\frac{1}{4},-\frac{1}{4},-\frac{15}{32}\right) \\\left(0,\frac{1}{2},-\frac{13}{32},-\frac{13}{32},-\frac{1}{4},\frac{1}{4},-\frac{1}{32}\right)\end{array}\)
\(\frac{1}{\sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}\left(\begin{array}{ccccccc} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & 0 \\ \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & 0 \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & 0 & 0 & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & 0 \\\end{array}\right)\)	\(\begin{array}{l}\left(0,\frac{1}{2},-\frac{5}{32},-\frac{5}{32},-\frac{1}{4},\frac{1}{4},\frac{7}{32}\right) \\\left(0,\frac{1}{2},-\frac{11}{32},-\frac{11}{32},\frac{1}{4},-\frac{1}{4},\frac{9}{32}\right) \\\left(0,\frac{1}{2},\frac{11}{32},\frac{11}{32},-\frac{1}{4},\frac{1}{4},-\frac{9}{32}\right) \\\left(0,\frac{1}{2},\frac{5}{32},\frac{5}{32},\frac{1}{4},-\frac{1}{4},-\frac{7}{32}\right)\end{array}\)

\(S\)-matrix

Twist factors

\(\frac{1}{\sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}\left(\begin{array}{ccccccc} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & 0 \\ \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & 0 \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & 0 & 0 & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & 0 \\\end{array}\right)\)

\(\begin{array}{l}\left(0,\frac{1}{2},\frac{13}{32},\frac{13}{32},\frac{1}{4},-\frac{1}{4},\frac{1}{32}\right) \\\left(0,\frac{1}{2},\frac{3}{32},\frac{3}{32},-\frac{1}{4},\frac{1}{4},\frac{15}{32}\right) \\\left(0,\frac{1}{2},-\frac{3}{32},-\frac{3}{32},\frac{1}{4},-\frac{1}{4},-\frac{15}{32}\right) \\\left(0,\frac{1}{2},-\frac{13}{32},-\frac{13}{32},-\frac{1}{4},\frac{1}{4},-\frac{1}{32}\right)\end{array}\)

\(\frac{1}{\sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}\left(\begin{array}{ccccccc} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,3\right] & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & 0 \\ \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & 0 \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & -\frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,4\right] & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \frac{1}{2 \sqrt{\frac{2}{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)}}} & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \text{Root}\left[128 x^4-32 x^2+1,2\right] & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} \\ \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & 0 & 0 & \frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & 0 \\\end{array}\right)\)

\(\begin{array}{l}\left(0,\frac{1}{2},-\frac{5}{32},-\frac{5}{32},-\frac{1}{4},\frac{1}{4},\frac{7}{32}\right) \\\left(0,\frac{1}{2},-\frac{11}{32},-\frac{11}{32},\frac{1}{4},-\frac{1}{4},\frac{9}{32}\right) \\\left(0,\frac{1}{2},\frac{11}{32},\frac{11}{32},-\frac{1}{4},\frac{1}{4},-\frac{9}{32}\right) \\\left(0,\frac{1}{2},\frac{5}{32},\frac{5}{32},\frac{1}{4},-\frac{1}{4},-\frac{7}{32}\right)\end{array}\)

Adjoint Subring

Particles \(\mathbf{1}, \mathbf{2}, \mathbf{5}, \mathbf{6}\), form the adjoint subring \(\text{PSU(2})_6:\ \text{FR}^{4,0}_{4}\) .

The upper central series is the following: \(\text{SU}(2)_6 \underset{ \mathbf{1}, \mathbf{2}, \mathbf{5}, \mathbf{6} }{\supset} \text{PSU(2})_6\)

Universal grading

Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{2}'\), 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

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