\(\text{FR}^{9,0}_{18}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{5} & \mathbf{6} & \mathbf{6} & \mathbf{5} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{5} & \mathbf{5} & \mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{7} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{3}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(2.89122\) | \(\text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,4\right]\) |
| \(\mathbf{6}\) | \(2.89122\) | \(\text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,4\right]\) |
| \(\mathbf{7}\) | \(3.46793\) | \(\text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,4\right]\) |
| \(\mathbf{8}\) | \(3.46793\) | \(\text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,4\right]\) |
| \(\mathbf{9}\) | \(3.6674\) | \(\text{Root}\left[x^4-3 x^3-8 x^2+16 x+16,4\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(58.2213\) | \(2 \text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,4\right]^2+2 \text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,4\right]^2+\text{Root}\left[x^4-3 x^3-8 x^2+16 x+16,4\right]^2+4\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & \text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,4\right] & \text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,4\right] & \text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,4\right] & \text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,4\right] & \text{Root}\left[x^4-3 x^3-8 x^2+16 x+16,4\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,3\right] & \text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,3\right] & \text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,2\right] & \text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,2\right] & \text{Root}\left[x^4-3 x^3-8 x^2+16 x+16,1\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,2\right] & \text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,2\right] & \text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,1\right] & \text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,1\right] & \text{Root}\left[x^4-3 x^3-8 x^2+16 x+16,3\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,1\right] & \text{Root}\left[x^4-3 x^3-2 x^2+6 x+2,1\right] & \text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,3\right] & \text{Root}\left[x^4-2 x^3-6 x^2+2 x+4,3\right] & \text{Root}\left[x^4-3 x^3-8 x^2+16 x+16,2\right] \\ 1 & -1 & -1 & 1 & \text{Root}\left[x^3-2 x^2-2 x+2,2\right] & \text{Root}\left[x^3+2 x^2-2 x-2,2\right] & \text{Root}\left[x^3-4 x-2,3\right] & \text{Root}\left[x^3-4 x+2,1\right] & 0 \\ 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & -1 & 1 & \text{Root}\left[x^3-2 x^2-2 x+2,3\right] & \text{Root}\left[x^3+2 x^2-2 x-2,1\right] & \text{Root}\left[x^3-4 x-2,1\right] & \text{Root}\left[x^3-4 x+2,3\right] & 0 \\ 1 & -1 & -1 & 1 & \text{Root}\left[x^3-2 x^2-2 x+2,1\right] & \text{Root}\left[x^3+2 x^2-2 x-2,3\right] & \text{Root}\left[x^3-4 x-2,2\right] & \text{Root}\left[x^3-4 x+2,2\right] & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.891 & 2.891 & 3.468 & 3.468 & 3.667 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.705 & 1.705 & -0.7989 & -0.7989 & -2.268 \\ 1.000 & 1.000 & 1.000 & 1.000 & -0.3174 & -0.3174 & -1.582 & -1.582 & 2.401 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.278 & -1.278 & 0.9128 & 0.9128 & -0.8012 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0.6889 & -0.6889 & 2.214 & -2.214 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 2.481 & -2.481 & -1.675 & 1.675 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & -1.170 & 1.170 & -0.5392 & 0.5392 & 0 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
This fusion ring does not provide any representations of $SL_2(\mathbb{Z}).$
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
This fusion ring has no categorifications because of the extended cyclotomic criterion.
Data
Download links for numeric data: