\(\text{FR}^{9,2}_{27}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{8} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{8} & \mathbf{9} \\ \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{7} & \mathbf{6} & \mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{6} & \mathbf{7} & \mathbf{6} & \mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{9} & \mathbf{9} & \mathbf{8} & \mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{8} & \mathbf{8} & \mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{6} \ \mathbf{7}), (\mathbf{8} \ \mathbf{9})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{3}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5}\}\) | \(\left.\text{Rep(}D_4\right):\ \text{FR}^{5,0}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(2.\) | \(2\) |
| \(\mathbf{6}\) | \(3.60388\) | \(\text{Root}\left[x^3-2 x^2-8 x+8,3\right]\) |
| \(\mathbf{7}\) | \(3.60388\) | \(\text{Root}\left[x^3-2 x^2-8 x+8,3\right]\) |
| \(\mathbf{8}\) | \(4.49396\) | \(\text{Root}\left[x^3-4 x^2-4 x+8,3\right]\) |
| \(\mathbf{9}\) | \(4.49396\) | \(\text{Root}\left[x^3-4 x^2-4 x+8,3\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(74.3672\) | \(2 \text{Root}\left[x^3-2 x^2-8 x+8,3\right]^2+2 \text{Root}\left[x^3-4 x^2-4 x+8,3\right]^2+8\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1 & 1 & 1 & 1 & 2 & \text{Root}\left[x^3-2 x^2-8 x+8,3\right] & \text{Root}\left[x^3-2 x^2-8 x+8,3\right] & \text{Root}\left[x^3-4 x^2-4 x+8,3\right] & \text{Root}\left[x^3-4 x^2-4 x+8,3\right] \\ 1 & 1 & 1 & 1 & 2 & \text{Root}\left[x^3-2 x^2-8 x+8,2\right] & \text{Root}\left[x^3-2 x^2-8 x+8,2\right] & \text{Root}\left[x^3-4 x^2-4 x+8,1\right] & \text{Root}\left[x^3-4 x^2-4 x+8,1\right] \\ 1 & 1 & 1 & 1 & 2 & \text{Root}\left[x^3-2 x^2-8 x+8,1\right] & \text{Root}\left[x^3-2 x^2-8 x+8,1\right] & \text{Root}\left[x^3-4 x^2-4 x+8,2\right] & \text{Root}\left[x^3-4 x^2-4 x+8,2\right] \\ 1 & 1 & 1 & 1 & -2 & 0 & 0 & 0 & 0 \\ 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & 0 & -\sqrt{2} & \sqrt{2} & 0 & 0 \\ 1 & -1 & 1 & -1 & 0 & \sqrt{2} & -\sqrt{2} & 0 & 0 \\ 1 & -1 & -1 & 1 & 0 & 0 & 0 & i \sqrt{2} & -i \sqrt{2} \\ 1 & -1 & -1 & 1 & 0 & 0 & 0 & -i \sqrt{2} & i \sqrt{2} \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & 3.604 & 3.604 & 4.494 & 4.494 \\ 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & 0.8901 & 0.8901 & -1.604 & -1.604 \\ 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & -2.494 & -2.494 & 1.110 & 1.110 \\ 1.000 & 1.000 & 1.000 & 1.000 & -2.000 & 0 & 0 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 & -1.414 & 1.414 & 0 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 & 1.414 & -1.414 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 & 0 & 1.414 i & -1.414 i \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 & 0 & -1.414 i & 1.414 i \\ \hline \end{array}\]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
Data
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