\(\mathbb{Z}_7:\ \text{FR}^{7,6}_{1}\)

Fusion Rules

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

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

\[\{(\mathbf{2} \ \mathbf{5} \ \mathbf{7} \ \mathbf{3} \ \mathbf{4} \ \mathbf{6}), (\mathbf{2} \ \mathbf{6} \ \mathbf{4} \ \mathbf{3} \ \mathbf{7} \ \mathbf{5})\}\]

Quantum Dimensions

Particle	Numeric	Symbolic
\(\mathbf{1}\)	\(1.\)	\(1\)
\(\mathbf{2}\)	\(1.\)	\(1\)
\(\mathbf{3}\)	\(1.\)	\(1\)
\(\mathbf{4}\)	\(1.\)	\(1\)
\(\mathbf{5}\)	\(1.\)	\(1\)
\(\mathbf{6}\)	\(1.\)	\(1\)
\(\mathbf{7}\)	\(1.\)	\(1\)
\(\mathcal{D}_{FP}^2\)	\(7.\)	\(7\)

Characters

The symbolic character table is the following

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

The numeric character table is the following

\[\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{3} & \mathbf{2} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 1.000 & 0.6235-0.7818 i & 0.6235+0.7818 i & -0.9010-0.4339 i & -0.9010+0.4339 i & -0.2225+0.9749 i & -0.2225-0.9749 i \\ 1.000 & 0.6235+0.7818 i & 0.6235-0.7818 i & -0.9010+0.4339 i & -0.9010-0.4339 i & -0.2225-0.9749 i & -0.2225+0.9749 i \\ 1.000 & -0.2225-0.9749 i & -0.2225+0.9749 i & 0.6235+0.7818 i & 0.6235-0.7818 i & -0.9010-0.4339 i & -0.9010+0.4339 i \\ 1.000 & -0.2225+0.9749 i & -0.2225-0.9749 i & 0.6235-0.7818 i & 0.6235+0.7818 i & -0.9010+0.4339 i & -0.9010-0.4339 i \\ 1.000 & -0.9010+0.4339 i & -0.9010-0.4339 i & -0.2225+0.9749 i & -0.2225-0.9749 i & 0.6235+0.7818 i & 0.6235-0.7818 i \\ 1.000 & -0.9010-0.4339 i & -0.9010+0.4339 i & -0.2225-0.9749 i & -0.2225+0.9749 i & 0.6235-0.7818 i & 0.6235+0.7818 i \\ \hline \end{array}\]

Modular Data

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

\(S\)-matrix	Twist factors
\(\frac{1}{\sqrt{7}}\left(\begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] \\\end{array}\right)\)	\(\begin{array}{l}\left(0,-\frac{3}{7},-\frac{3}{7},\frac{1}{7},\frac{1}{7},\frac{2}{7},\frac{2}{7}\right)\end{array}\)
\(\frac{1}{\sqrt{7}}\left(\begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] \\\end{array}\right)\)	\(\begin{array}{l}\left(0,\frac{3}{7},\frac{3}{7},-\frac{1}{7},-\frac{1}{7},-\frac{2}{7},-\frac{2}{7}\right)\end{array}\)

\(S\)-matrix

Twist factors

\(\frac{1}{\sqrt{7}}\left(\begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] \\\end{array}\right)\)

\(\begin{array}{l}\left(0,-\frac{3}{7},-\frac{3}{7},\frac{1}{7},\frac{1}{7},\frac{2}{7},\frac{2}{7}\right)\end{array}\)

\(\frac{1}{\sqrt{7}}\left(\begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] \\ 1 & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,6\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,5\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,10\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,9\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,2\right] & \sqrt{7} \text{Root}\left[117649 x^{12}+16807 x^{10}+2401 x^8+343 x^6+49 x^4+7 x^2+1,1\right] \\\end{array}\right)\)

\(\begin{array}{l}\left(0,\frac{3}{7},\frac{3}{7},-\frac{1}{7},-\frac{1}{7},-\frac{2}{7},-\frac{2}{7}\right)\end{array}\)

Adjoint Subring

The adjoint subring is the trivial ring.

The upper central series is the following: \(\mathbb{Z}_7 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)

Universal grading

Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}', \text{deg}(\mathbf{4}) = \mathbf{4}', \text{deg}(\mathbf{5}) = \mathbf{5}', \text{deg}(\mathbf{6}) = \mathbf{6}', \text{deg}(\mathbf{7}) = \mathbf{7}'\), where the degrees form the group \(\mathbb{Z}_7\) with multiplication table:

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

Categorifications

Data

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