FR117,2\text{FR}^{7,2}_{11}

Fusion Rules

12345672143657342+51+64+73+75+6431+62+53+74+75+6564+73+71+5+62+5+63+4+7653+74+72+5+61+5+63+4+7775+65+63+4+73+4+71+2+5+6\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{2}+\mathbf{5} & \mathbf{1}+\mathbf{6} & \mathbf{4}+\mathbf{7} & \mathbf{3}+\mathbf{7} & \mathbf{5}+\mathbf{6} \\ \mathbf{4} & \mathbf{3} & \mathbf{1}+\mathbf{6} & \mathbf{2}+\mathbf{5} & \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:

{(3 4)}\{(\mathbf{3} \ \mathbf{4})\}

The following elements form non-trivial sub fusion rings

Elements SubRing
{1,2}\{\mathbf{1},\mathbf{2}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,2,5,6}\{\mathbf{1},\mathbf{2},\mathbf{5},\mathbf{6}\} PSU(2)6: FR44,0\text{PSU(2})_6:\ \text{FR}^{4,0}_{4}

Frobenius-Perron Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 1.847761.84776 2+2\sqrt{2+\sqrt{2}}
4\mathbf{4} 1.847761.84776 2+2\sqrt{2+\sqrt{2}}
5\mathbf{5} 2.414212.41421 1+21+\sqrt{2}
6\mathbf{6} 2.414212.41421 1+21+\sqrt{2}
7\mathbf{7} 2.613132.61313 2(2+2)\sqrt{2 \left(2+\sqrt{2}\right)}
DFP2\mathcal{D}_{FP}^2 27.313727.3137 6+22+2(1+2)2+2(2+2)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

124356711Root[x44x2+2,4]Root[x44x2+2,4]1+21+2Root[x48x2+8,4]11Root[x44x2+2,3]Root[x44x2+2,3]1212Root[x48x2+8,2]11Root[x44x2+2,2]Root[x44x2+2,2]1212Root[x48x2+8,3]11Root[x44x2+2,1]Root[x44x2+2,1]1+21+2Root[x48x2+8,1]110011011i2i211011i2i2110\begin{array}{|ccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \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 & 0 & 0 & 1 & -1 & 0 \\ 1 & -1 & i \sqrt{2} & -i \sqrt{2} & -1 & 1 & 0 \\ 1 & -1 & -i \sqrt{2} & i \sqrt{2} & -1 & 1 & 0 \\ \hline \end{array}

The numeric character table is the following

12435671.0001.0001.8481.8482.4142.4142.6131.0001.0000.76540.76540.41420.41421.0821.0001.0000.76540.76540.41420.41421.0821.0001.0001.8481.8482.4142.4142.6131.0001.000001.0001.00001.0001.0001.414i1.414i1.0001.00001.0001.0001.414i1.414i1.0001.0000\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \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 & 0 & 0 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & 1.414 i & -1.414 i & -1.000 & 1.000 & 0 \\ 1.000 & -1.000 & -1.414 i & 1.414 i & -1.000 & 1.000 & 0 \\ \hline \end{array}

Representations of SL2(Z)SL_2(\mathbb{Z})

The matching SS-matrices and twist factors are the following

SS-matrix Twist factors
16+22+2(1+2)2+2(2+2)(6+22+2(1+2)2+2(2+2)Root[128x432x2+1,3]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,3]1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]126+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,3]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,3]1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]126+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)12i6+22+2(1+2)2+2(2+2)12i6+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)01226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)12i6+22+2(1+2)2+2(2+2)12i6+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)06+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,2]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,2]126+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,2]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,2]126+22+2(1+2)2+2(2+2)126+22+2(1+2)2+2(2+2)126+22+2(1+2)2+2(2+2)00126+22+2(1+2)2+2(2+2)126+22+2(1+2)2+2(2+2)0)\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} i \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & \frac{1}{2} i \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} i \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} i \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) (0,12,132,132,14,14,1132)(0,12,732,732,14,14,1332)(0,12,1532,1532,14,14,532)(0,12,932,932,14,14,332)\begin{array}{l}\left(0,\frac{1}{2},\frac{1}{32},\frac{1}{32},\frac{1}{4},-\frac{1}{4},-\frac{11}{32}\right) \\\left(0,\frac{1}{2},\frac{7}{32},\frac{7}{32},-\frac{1}{4},\frac{1}{4},-\frac{13}{32}\right) \\\left(0,\frac{1}{2},-\frac{15}{32},-\frac{15}{32},\frac{1}{4},-\frac{1}{4},\frac{5}{32}\right) \\\left(0,\frac{1}{2},-\frac{9}{32},-\frac{9}{32},-\frac{1}{4},\frac{1}{4},\frac{3}{32}\right)\end{array}
16+22+2(1+2)2+2(2+2)(6+22+2(1+2)2+2(2+2)Root[128x432x2+1,3]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,3]1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]126+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,3]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,3]1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]126+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)12i6+22+2(1+2)2+2(2+2)12i6+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)01226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)12i6+22+2(1+2)2+2(2+2)12i6+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)06+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,2]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,2]126+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,4]1226+22+2(1+2)2+2(2+2)1226+22+2(1+2)2+2(2+2)6+22+2(1+2)2+2(2+2)Root[128x432x2+1,2]6+22+2(1+2)2+2(2+2)Root[128x432x2+1,2]126+22+2(1+2)2+2(2+2)126+22+2(1+2)2+2(2+2)126+22+2(1+2)2+2(2+2)00126+22+2(1+2)2+2(2+2)126+22+2(1+2)2+2(2+2)0)\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} i \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & -\frac{1}{2} i \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} i \sqrt{6+2 \sqrt{2}+2 \left(1+\sqrt{2}\right)^2+2 \left(2+\sqrt{2}\right)} & \frac{1}{2} i \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) (0,12,932,932,14,14,332)(0,12,1532,1532,14,14,532)(0,12,732,732,14,14,1332)(0,12,132,132,14,14,1132)\begin{array}{l}\left(0,\frac{1}{2},\frac{9}{32},\frac{9}{32},\frac{1}{4},-\frac{1}{4},-\frac{3}{32}\right) \\\left(0,\frac{1}{2},\frac{15}{32},\frac{15}{32},-\frac{1}{4},\frac{1}{4},-\frac{5}{32}\right) \\\left(0,\frac{1}{2},-\frac{7}{32},-\frac{7}{32},\frac{1}{4},-\frac{1}{4},\frac{13}{32}\right) \\\left(0,\frac{1}{2},-\frac{1}{32},-\frac{1}{32},-\frac{1}{4},\frac{1}{4},\frac{11}{32}\right)\end{array}

Adjoint Subring

Elements 1,2,5,6\mathbf{1}, \mathbf{2}, \mathbf{5}, \mathbf{6}, form the adjoint subring PSU(2)6: FR44,0\text{PSU(2})_6:\ \text{FR}^{4,0}_{4} .

The upper central series is the following: FR117,21,2,5,6PSU(2)6\text{FR}^{7,2}_{11} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{5}, \mathbf{6} }{\supset} \text{PSU(2})_6

Universal grading

Each particle can be graded as follows: deg(1)=1,deg(2)=1,deg(3)=2,deg(4)=2,deg(5)=1,deg(6)=1,deg(7)=2\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 Z2\mathbb{Z}_2 with multiplication table:

1221\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}

Categorifications

Data

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