\(\text{FR}^{7,4}_{7}\)
Fusion Rules
\[\begin{array}{|lllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{4} & \mathbf{6} & \mathbf{7} & \mathbf{5} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{7} & \mathbf{5} & \mathbf{6} \\ \mathbf{4} & \mathbf{4} & \mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{7} & \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}) (\mathbf{6} \ \mathbf{7})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(3.94282\) | \(\text{Root}\left[x^3-3 x^2-6 x+9,3\right]\) |
| \(\mathbf{5}\) | \(4.18194\) | \(\text{Root}\left[x^3-4 x^2-x+1,3\right]\) |
| \(\mathbf{6}\) | \(4.18194\) | \(\text{Root}\left[x^3-4 x^2-x+1,3\right]\) |
| \(\mathbf{7}\) | \(4.18194\) | \(\text{Root}\left[x^3-4 x^2-x+1,3\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(71.0118\) | \(\text{Root}\left[x^3-3 x^2-6 x+9,3\right]^2+3 \text{Root}\left[x^3-4 x^2-x+1,3\right]^2+3\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \hline 1 & e^{-\frac{2 \pi}{3} i } & e^{\frac{2 \pi}{3} i } & 0 & 1 & e^{-\frac{2 \pi}{3} i } & e^{\frac{2 \pi}{3} i } \\ 1 & e^{-\frac{2 \pi}{3} i } & e^{\frac{2 \pi}{3} i } & 0 & -1 & e^{\frac{ \pi}{3} i } & e^{-\frac{ \pi}{3} i } \\ 1 & e^{\frac{2 \pi}{3} i } & e^{-\frac{2 \pi}{3} i } & 0 & -1 & e^{-\frac{ \pi}{3} i } & e^{\frac{ \pi}{3} i }\\ 1 & e^{\frac{2 \pi}{3} i } & e^{-\frac{2 \pi}{3} i } & 0 & 1 & e^{\frac{2 \pi}{3} i } & e^{-\frac{2 \pi}{3} i } \\ 1 & 1 & 1 & s_2 & r_1 & r_1 & r_1 \\ 1 & 1 & 1 & s_1 & r_2 & r_2 & r_2 \\ 1 & 1 & 1 & s_3 & r_3 & r_3 & r_3 \\ \hline \end{array}\]Where $r_i$ is the i’th root of the polynomial $ 1 - x - 4 x^2 + x^3$ and $s_i$ is the i’th root of the polynomial $9 -6x -3x^2+x^3$.
The numeric character table is the following
\[\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \hline 1. & -0.5-0.866025 i & -0.5+0.866025 i & 0. & 1. & -0.5-0.866025 i & -0.5+0.866025 i \\ 1. & -0.5-0.866025 i & -0.5+0.866025 i & 0. & -1. & 0.5 +0.866025 i & 0.5 -0.866025 i \\ 1. & -0.5+0.866025 i & -0.5-0.866025 i & 0. & -1. & 0.5 -0.866025 i & 0.5 +0.866025 i \\ 1. & -0.5+0.866025 i & -0.5-0.866025 i & 0. & 1. & -0.5+0.866025 i & -0.5-0.866025 i \\ 1. & 1. & 1. & 1.11126 & -0.588364 & -0.588364 & -0.588364 \\ 1. & 1. & 1. & -2.05408 & 0.406421 & 0.406421 & 0.406421 \\ 1. & 1. & 1. & 3.94282 & 4.18194 & 4.18194 & 4.18194 \\ \hline \end{array}\]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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