\(\text{FR}^{7,4}_{3}\)

Fusion Rules

\[\begin{array}{|lllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{6} & \mathbf{7} & \mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{5} \\ \mathbf{7} & \mathbf{7} & \mathbf{6} & \mathbf{6} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{5} \\ \hline \end{array}\]

The fusion rules are invariant under the group generated by the following permutations:

\[\{(\mathbf{3} \ \mathbf{4}), (\mathbf{6} \ \mathbf{7})\}\]

The following particles form non-trivial sub fusion rings

Particles SubRing
\(\{\mathbf{1},\mathbf{2}\}\) \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\)
\(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) \(\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}\)
\(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5}\}\) \(\left.\text{TY(}\mathbb{Z}_4\right):\ \text{FR}^{5,2}_{1}\)

Quantum Dimensions

Particle Numeric Symbolic
\(\mathbf{1}\) \(1.\) \(1\)
\(\mathbf{2}\) \(1.\) \(1\)
\(\mathbf{3}\) \(1.\) \(1\)
\(\mathbf{4}\) \(1.\) \(1\)
\(\mathbf{5}\) \(2.\) \(2\)
\(\mathbf{6}\) \(2.\) \(2\)
\(\mathbf{7}\) \(2.\) \(2\)
\(\mathcal{D}_{FP}^2\) \(16.\) \(16\)

Characters

The symbolic character table is the following

\[\begin{array}{|ccccccc|} \hline \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \hline 1 & 1 & 1 & 1 & 2 & 2 & 2 \\ 1 & 1 & 1 & 1 & 2 & -2 & -2 \\ 1 & 1 & 1 & 1 & -2 & 0 & 0 \\ 1 & i & -i & -1 & 0 & 0 & 0 \\ 1 & -i & i & -1 & 0 & 0 & 0 \\ 1 & -1 & -1 & 1 & 0 & i \sqrt{2} & -i \sqrt{2} \\ 1 & -1 & -1 & 1 & 0 & -i \sqrt{2} & i \sqrt{2} \\ \hline \end{array}\]

The numeric character table is the following

\[\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & 2.000 & 2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & -2.000 & -2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -2.000 & 0 & 0 \\ 1.000 & 1.000 i & -1.000 i & -1.000 & 0 & 0 & 0 \\ 1.000 & -1.000 i & 1.000 i & -1.000 & 0 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 1.414 i & -1.414 i \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & -1.414 i & 1.414 i \\ \hline \end{array}\]

Modular Data

This fusion ring does not have any matching \(S\)-and \(T\)-matrices.

Adjoint Subring

Particles \(\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}\), form the adjoint subring \(\left.\text{TY(}\mathbb{Z}_4\right):\ \text{FR}^{5,2}_{1}\) .

The upper central series is the following: \(\text{FR}^{7,4}_{3} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5} }{\supset} \left.\text{TY(}\mathbb{Z}_4\right) \underset{ \mathbf{1}, \mathbf{2}, \mathbf{4}, \mathbf{5} }{\supset} \mathbb{Z}_4 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)

Universal grading

Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{2}'\), where the degrees form the group \(\mathbb{Z}_2\) with multiplication table:

\[\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}\]

Categorifications

Data

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