\(\text{FR}^{9,6}_{14}\)
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{5} & \mathbf{9} & \mathbf{8} & \mathbf{7} & \mathbf{6} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{7} & \mathbf{9} & \mathbf{6} & \mathbf{8} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{8} & \mathbf{6} & \mathbf{9} & \mathbf{7} \\ \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{9} & \mathbf{7} & \mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{6} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{6} \ \mathbf{9}) (\mathbf{7} \ \mathbf{8}), (\mathbf{3} \ \mathbf{4}) (\mathbf{6} \ \mathbf{7}) (\mathbf{8} \ \mathbf{9})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(4.96239\) | \(\text{Root}\left[x^3-4 x^2-8 x+16,3\right]\) |
| \(\mathbf{6}\) | \(5.15633\) | \(\text{Root}\left[x^3-5 x^2-x+1,3\right]\) |
| \(\mathbf{7}\) | \(5.15633\) | \(\text{Root}\left[x^3-5 x^2-x+1,3\right]\) |
| \(\mathbf{8}\) | \(5.15633\) | \(\text{Root}\left[x^3-5 x^2-x+1,3\right]\) |
| \(\mathbf{9}\) | \(5.15633\) | \(\text{Root}\left[x^3-5 x^2-x+1,3\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(134.976\) | \(\text{Root}\left[x^3-4 x^2-8 x+16,3\right]^2+4 \text{Root}\left[x^3-5 x^2-x+1,3\right]^2+4\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{8} \\ \hline 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-8 x+16,3\right] & \text{Root}\left[x^3-5 x^2-x+1,3\right] & \text{Root}\left[x^3-5 x^2-x+1,3\right] & \text{Root}\left[x^3-5 x^2-x+1,3\right] & \text{Root}\left[x^3-5 x^2-x+1,3\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-8 x+16,2\right] & \text{Root}\left[x^3-5 x^2-x+1,1\right] & \text{Root}\left[x^3-5 x^2-x+1,1\right] & \text{Root}\left[x^3-5 x^2-x+1,1\right] & \text{Root}\left[x^3-5 x^2-x+1,1\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-8 x+16,1\right] & \text{Root}\left[x^3-5 x^2-x+1,2\right] & \text{Root}\left[x^3-5 x^2-x+1,2\right] & \text{Root}\left[x^3-5 x^2-x+1,2\right] & \text{Root}\left[x^3-5 x^2-x+1,2\right] \\ 1 & 1 & -1 & -1 & 0 & -i & i & i & -i \\ 1 & 1 & -1 & -1 & 0 & i & -i & -i & i \\ 1 & -1 & -i & i & 0 & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] \\ 1 & -1 & i & -i & 0 & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] \\ 1 & -1 & i & -i & 0 & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] \\ 1 & -1 & -i & i & 0 & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 4.962 & 5.156 & 5.156 & 5.156 & 5.156 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.378 & -0.5254 & -0.5254 & -0.5254 & -0.5254 \\ 1.000 & 1.000 & 1.000 & 1.000 & -2.340 & 0.3691 & 0.3691 & 0.3691 & 0.3691 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & -1.000 i & 1.000 i & 1.000 i & -1.000 i \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 1.000 i & -1.000 i & -1.000 i & 1.000 i \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 0 & 0.7071+0.7071 i & 0.7071-0.7071 i & -0.7071+0.7071 i & -0.7071-0.7071 i \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 0 & 0.7071-0.7071 i & 0.7071+0.7071 i & -0.7071-0.7071 i & -0.7071+0.7071 i \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 0 & -0.7071+0.7071 i & -0.7071-0.7071 i & 0.7071+0.7071 i & 0.7071-0.7071 i \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 0 & -0.7071-0.7071 i & -0.7071+0.7071 i & 0.7071-0.7071 i & 0.7071+0.7071 i \\ \hline \end{array}\]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
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