\(\text{FR}^{9,0}_{33}\)
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{3} & \mathbf{5} & \mathbf{4} & \mathbf{9} & \mathbf{8} & \mathbf{7} & \mathbf{6} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{4}+\mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{6}+\mathbf{9} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{9} \\ \mathbf{4} & \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{4} & \mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{9} & \mathbf{6}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7} \\ \mathbf{7} & \mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{8} & \mathbf{7} & \mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{6} & \mathbf{6}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5}\}\) | \(\left.\text{Rep(}S_4\right):\ \text{FR}^{5,0}_{6}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.\) | \(2\) |
| \(\mathbf{4}\) | \(3.\) | \(3\) |
| \(\mathbf{5}\) | \(3.\) | \(3\) |
| \(\mathbf{6}\) | \(3.64575\) | \(1+\sqrt{7}\) |
| \(\mathbf{7}\) | \(3.64575\) | \(1+\sqrt{7}\) |
| \(\mathbf{8}\) | \(3.64575\) | \(1+\sqrt{7}\) |
| \(\mathbf{9}\) | \(3.64575\) | \(1+\sqrt{7}\) |
| \(\mathcal{D}_{FP}^2\) | \(77.166\) | \(24+4 \left(1+\sqrt{7}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & 2 & 3 & 3 & 1+\sqrt{7} & 1+\sqrt{7} & 1+\sqrt{7} & 1+\sqrt{7} \\ 1 & 1 & 2 & 3 & 3 & 1-\sqrt{7} & 1-\sqrt{7} & 1-\sqrt{7} & 1-\sqrt{7} \\ 1 & 1 & 2 & -1 & -1 & -\sqrt{2} & \sqrt{2} & \sqrt{2} & -\sqrt{2} \\ 1 & 1 & 2 & -1 & -1 & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & \sqrt{2} \\ 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 1 & -1 & \text{Root}\left[x^3+2 x^2-2 x-2,2\right] & \text{Root}\left[x^3-4 x-2,3\right] & \text{Root}\left[x^3-4 x+2,1\right] & \text{Root}\left[x^3-2 x^2-2 x+2,2\right] \\ 1 & -1 & 0 & 1 & -1 & \text{Root}\left[x^3+2 x^2-2 x-2,1\right] & \text{Root}\left[x^3-4 x-2,1\right] & \text{Root}\left[x^3-4 x+2,3\right] & \text{Root}\left[x^3-2 x^2-2 x+2,3\right] \\ 1 & -1 & 0 & 1 & -1 & \text{Root}\left[x^3+2 x^2-2 x-2,3\right] & \text{Root}\left[x^3-4 x-2,2\right] & \text{Root}\left[x^3-4 x+2,2\right] & \text{Root}\left[x^3-2 x^2-2 x+2,1\right] \\ 1 & -1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 2.000 & 3.000 & 3.000 & 3.646 & 3.646 & 3.646 & 3.646 \\ 1.000 & 1.000 & 2.000 & 3.000 & 3.000 & -1.646 & -1.646 & -1.646 & -1.646 \\ 1.000 & 1.000 & 2.000 & -1.000 & -1.000 & -1.414 & 1.414 & 1.414 & -1.414 \\ 1.000 & 1.000 & 2.000 & -1.000 & -1.000 & 1.414 & -1.414 & -1.414 & 1.414 \\ 1.000 & 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & 0 & 1.000 & -1.000 & -0.6889 & 2.214 & -2.214 & 0.6889 \\ 1.000 & -1.000 & 0 & 1.000 & -1.000 & -2.481 & -1.675 & 1.675 & 2.481 \\ 1.000 & -1.000 & 0 & 1.000 & -1.000 & 1.170 & -0.5392 & 0.5392 & -1.170 \\ 1.000 & -1.000 & 0 & -1.000 & 1.000 & 0 & 0 & 0 & 0 \\ \hline \end{array}\]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
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