\(\text{FR}^{9,4}_{27}\)
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{4} & \mathbf{5} & \mathbf{9} & \mathbf{7} & \mathbf{8} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{8} & \mathbf{9} & \mathbf{7} \\ \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{1}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} \\ \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{3}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{2}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{9} & \mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{7} & \mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{7} \ \mathbf{8} \ \mathbf{9}), (\mathbf{7} \ \mathbf{9} \ \mathbf{8}), (\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6}) (\mathbf{7} \ \mathbf{8})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(3.65859\) | \(\text{Root}\left[x^3-x^2-10 x+1,3\right]\) |
| \(\mathbf{5}\) | \(3.65859\) | \(\text{Root}\left[x^3-x^2-10 x+1,3\right]\) |
| \(\mathbf{6}\) | \(3.65859\) | \(\text{Root}\left[x^3-x^2-10 x+1,3\right]\) |
| \(\mathbf{7}\) | \(4.12842\) | \(\text{Root}\left[x^3-6 x^2+7 x+3,3\right]\) |
| \(\mathbf{8}\) | \(4.12842\) | \(\text{Root}\left[x^3-6 x^2+7 x+3,3\right]\) |
| \(\mathbf{9}\) | \(4.12842\) | \(\text{Root}\left[x^3-6 x^2+7 x+3,3\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(94.2873\) | \(3 \text{Root}\left[x^3-x^2-10 x+1,3\right]^2+3 \text{Root}\left[x^3-6 x^2+7 x+3,3\right]^2+3\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & 1 & \text{Root}\left[x^3-x^2-10 x+1,3\right] & \text{Root}\left[x^3-x^2-10 x+1,3\right] & \text{Root}\left[x^3-x^2-10 x+1,3\right] & \text{Root}\left[x^3-6 x^2+7 x+3,3\right] & \text{Root}\left[x^3-6 x^2+7 x+3,3\right] & \text{Root}\left[x^3-6 x^2+7 x+3,3\right] \\ 1 & 1 & 1 & \text{Root}\left[x^3-x^2-10 x+1,2\right] & \text{Root}\left[x^3-x^2-10 x+1,2\right] & \text{Root}\left[x^3-x^2-10 x+1,2\right] & \text{Root}\left[x^3-6 x^2+7 x+3,1\right] & \text{Root}\left[x^3-6 x^2+7 x+3,1\right] & \text{Root}\left[x^3-6 x^2+7 x+3,1\right] \\ 1 & 1 & 1 & \text{Root}\left[x^3-x^2-10 x+1,1\right] & \text{Root}\left[x^3-x^2-10 x+1,1\right] & \text{Root}\left[x^3-x^2-10 x+1,1\right] & \text{Root}\left[x^3-6 x^2+7 x+3,2\right] & \text{Root}\left[x^3-6 x^2+7 x+3,2\right] & \text{Root}\left[x^3-6 x^2+7 x+3,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(1-i \sqrt{3}\right) & \frac{1}{2} \left(1+i \sqrt{3}\right) & -1 & 0 & 0 & 0 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(1+i \sqrt{3}\right) & \frac{1}{2} \left(1-i \sqrt{3}\right) & -1 & 0 & 0 & 0 \\ 2 & -1 & -1 & -1 & -1 & 2 & 0 & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 3.659 & 3.659 & 3.659 & 4.128 & 4.128 & 4.128 \\ 1.000 & 1.000 & 1.000 & 0.09911 & 0.09911 & 0.09911 & -0.3301 & -0.3301 & -0.3301 \\ 1.000 & 1.000 & 1.000 & -2.758 & -2.758 & -2.758 & 2.202 & 2.202 & 2.202 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.5000-0.8660 i & 0.5000+0.8660 i & -1.000 & 0 & 0 & 0 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.5000+0.8660 i & 0.5000-0.8660 i & -1.000 & 0 & 0 & 0 \\ 2.000 & -1.000 & -1.000 & -1.000 & -1.000 & 2.000 & 0 & 0 & 0 \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
The adjoint subring is the ring itself.
The upper central series is trivial.
Universal grading
This fusion ring allows only the trivial grading.
Categorifications
Data
Download links for numeric data: