\(\mathbb{Z}_9:\ \text{FR}^{9,8}_{1}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{6} & \mathbf{8} & \mathbf{9} & \mathbf{5} & \mathbf{7} & \mathbf{4} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{9} & \mathbf{7} & \mathbf{4} & \mathbf{8} & \mathbf{5} & \mathbf{6} \\ \mathbf{4} & \mathbf{6} & \mathbf{9} & \mathbf{8} & \mathbf{1} & \mathbf{7} & \mathbf{3} & \mathbf{2} & \mathbf{5} \\ \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{1} & \mathbf{9} & \mathbf{2} & \mathbf{6} & \mathbf{4} & \mathbf{3} \\ \mathbf{6} & \mathbf{9} & \mathbf{4} & \mathbf{7} & \mathbf{2} & \mathbf{5} & \mathbf{1} & \mathbf{3} & \mathbf{8} \\ \mathbf{7} & \mathbf{5} & \mathbf{8} & \mathbf{3} & \mathbf{6} & \mathbf{1} & \mathbf{4} & \mathbf{9} & \mathbf{2} \\ \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{9} & \mathbf{6} & \mathbf{1} \\ \mathbf{9} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{3} & \mathbf{8} & \mathbf{2} & \mathbf{1} & \mathbf{7} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}) (\mathbf{4} \ \mathbf{7} \ \mathbf{9} \ \mathbf{5} \ \mathbf{6} \ \mathbf{8}), (\mathbf{2} \ \mathbf{3}) (\mathbf{4} \ \mathbf{8} \ \mathbf{6} \ \mathbf{5} \ \mathbf{9} \ \mathbf{7})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(1.\) | \(1\) |
| \(\mathbf{6}\) | \(1.\) | \(1\) |
| \(\mathbf{7}\) | \(1.\) | \(1\) |
| \(\mathbf{8}\) | \(1.\) | \(1\) |
| \(\mathbf{9}\) | \(1.\) | \(1\) |
| \(\mathcal{D}_{FP}^2\) | \(9.\) | \(9\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{6} & \mathbf{7} & \mathbf{2} & \mathbf{3} & \mathbf{8} & \mathbf{9} & \mathbf{4} & \mathbf{5} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & \text{Root}\left[x^6+x^3+1,3\right] & \text{Root}\left[x^6+x^3+1,4\right] & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,5\right] & \text{Root}\left[x^6+x^3+1,6\right] & \text{Root}\left[x^6+x^3+1,2\right] & \text{Root}\left[x^6+x^3+1,1\right] \\ 1 & \text{Root}\left[x^6+x^3+1,4\right] & \text{Root}\left[x^6+x^3+1,3\right] & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,6\right] & \text{Root}\left[x^6+x^3+1,5\right] & \text{Root}\left[x^6+x^3+1,1\right] & \text{Root}\left[x^6+x^3+1,2\right] \\ 1 & \text{Root}\left[x^6+x^3+1,5\right] & \text{Root}\left[x^6+x^3+1,6\right] & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,2\right] & \text{Root}\left[x^6+x^3+1,1\right] & \text{Root}\left[x^6+x^3+1,4\right] & \text{Root}\left[x^6+x^3+1,3\right] \\ 1 & \text{Root}\left[x^6+x^3+1,6\right] & \text{Root}\left[x^6+x^3+1,5\right] & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,1\right] & \text{Root}\left[x^6+x^3+1,2\right] & \text{Root}\left[x^6+x^3+1,3\right] & \text{Root}\left[x^6+x^3+1,4\right] \\ 1 & \text{Root}\left[x^6+x^3+1,2\right] & \text{Root}\left[x^6+x^3+1,1\right] & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,4\right] & \text{Root}\left[x^6+x^3+1,3\right] & \text{Root}\left[x^6+x^3+1,6\right] & \text{Root}\left[x^6+x^3+1,5\right] \\ 1 & \text{Root}\left[x^6+x^3+1,1\right] & \text{Root}\left[x^6+x^3+1,2\right] & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,3\right] & \text{Root}\left[x^6+x^3+1,4\right] & \text{Root}\left[x^6+x^3+1,5\right] & \text{Root}\left[x^6+x^3+1,6\right] \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{6} & \mathbf{7} & \mathbf{2} & \mathbf{3} & \mathbf{8} & \mathbf{9} & \mathbf{4} & \mathbf{5} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 1.000 & 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.5000+0.8660 i & -0.5000-0.8660 i \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 1.000 & 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.5000-0.8660 i & -0.5000+0.8660 i \\ 1.000 & 0.1736-0.9848 i & 0.1736+0.9848 i & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.7660-0.6428 i & 0.7660+0.6428 i & -0.9397+0.3420 i & -0.9397-0.3420 i \\ 1.000 & 0.1736+0.9848 i & 0.1736-0.9848 i & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.7660+0.6428 i & 0.7660-0.6428 i & -0.9397-0.3420 i & -0.9397+0.3420 i \\ 1.000 & 0.7660-0.6428 i & 0.7660+0.6428 i & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.9397+0.3420 i & -0.9397-0.3420 i & 0.1736+0.9848 i & 0.1736-0.9848 i \\ 1.000 & 0.7660+0.6428 i & 0.7660-0.6428 i & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.9397-0.3420 i & -0.9397+0.3420 i & 0.1736-0.9848 i & 0.1736+0.9848 i \\ 1.000 & -0.9397+0.3420 i & -0.9397-0.3420 i & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.1736+0.9848 i & 0.1736-0.9848 i & 0.7660+0.6428 i & 0.7660-0.6428 i \\ 1.000 & -0.9397-0.3420 i & -0.9397+0.3420 i & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.1736-0.9848 i & 0.1736+0.9848 i & 0.7660-0.6428 i & 0.7660+0.6428 i \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,0,-\frac{4}{9},-\frac{4}{9},-\frac{1}{9},-\frac{1}{9},\frac{2}{9},\frac{2}{9}\right)\end{array}\) |
| \(\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,0,-\frac{1}{9},-\frac{1}{9},\frac{2}{9},\frac{2}{9},-\frac{4}{9},-\frac{4}{9}\right)\end{array}\) |
| \(\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,0,\frac{2}{9},\frac{2}{9},-\frac{4}{9},-\frac{4}{9},-\frac{1}{9},-\frac{1}{9}\right)\end{array}\) |
| \(\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,0,\frac{4}{9},\frac{4}{9},\frac{1}{9},\frac{1}{9},-\frac{2}{9},-\frac{2}{9}\right)\end{array}\) |
| \(\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,0,\frac{1}{9},\frac{1}{9},-\frac{2}{9},-\frac{2}{9},\frac{4}{9},\frac{4}{9}\right)\end{array}\) |
| \(\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\\end{array}\right)\) | \(\begin{array}{l}\left(0,0,0,-\frac{2}{9},-\frac{2}{9},\frac{4}{9},\frac{4}{9},\frac{1}{9},\frac{1}{9}\right)\end{array}\) |
Adjoint Subring
The adjoint subring is the trivial ring.
The upper central series is the following: \(\mathbb{Z}_9 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}', \text{deg}(\mathbf{4}) = \mathbf{4}', \text{deg}(\mathbf{5}) = \mathbf{5}', \text{deg}(\mathbf{6}) = \mathbf{6}', \text{deg}(\mathbf{7}) = \mathbf{7}', \text{deg}(\mathbf{8}) = \mathbf{8}', \text{deg}(\mathbf{9}) = \mathbf{9}'\), where the degrees form the group \(\mathbb{Z}_9\) with multiplication table:
\[\begin{array}{|lllllllll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' & \mathbf{5}' & \mathbf{6}' & \mathbf{7}' & \mathbf{8}' & \mathbf{9}' \\ \mathbf{2}' & \mathbf{3}' & \mathbf{1}' & \mathbf{6}' & \mathbf{8}' & \mathbf{9}' & \mathbf{5}' & \mathbf{7}' & \mathbf{4}' \\ \mathbf{3}' & \mathbf{1}' & \mathbf{2}' & \mathbf{9}' & \mathbf{7}' & \mathbf{4}' & \mathbf{8}' & \mathbf{5}' & \mathbf{6}' \\ \mathbf{4}' & \mathbf{6}' & \mathbf{9}' & \mathbf{8}' & \mathbf{1}' & \mathbf{7}' & \mathbf{3}' & \mathbf{2}' & \mathbf{5}' \\ \mathbf{5}' & \mathbf{8}' & \mathbf{7}' & \mathbf{1}' & \mathbf{9}' & \mathbf{2}' & \mathbf{6}' & \mathbf{4}' & \mathbf{3}' \\ \mathbf{6}' & \mathbf{9}' & \mathbf{4}' & \mathbf{7}' & \mathbf{2}' & \mathbf{5}' & \mathbf{1}' & \mathbf{3}' & \mathbf{8}' \\ \mathbf{7}' & \mathbf{5}' & \mathbf{8}' & \mathbf{3}' & \mathbf{6}' & \mathbf{1}' & \mathbf{4}' & \mathbf{9}' & \mathbf{2}' \\ \mathbf{8}' & \mathbf{7}' & \mathbf{5}' & \mathbf{2}' & \mathbf{4}' & \mathbf{3}' & \mathbf{9}' & \mathbf{6}' & \mathbf{1}' \\ \mathbf{9}' & \mathbf{4}' & \mathbf{6}' & \mathbf{5}' & \mathbf{3}' & \mathbf{8}' & \mathbf{2}' & \mathbf{1}' & \mathbf{7}' \\ \hline \end{array}\]Categorifications
Data
Download links for numeric data: