\(\left.\text{TY(}\mathbb{Z}_4\right):\ \text{FR}^{5,2}_{1}\)

Fusion Rules

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

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

\[\{(\mathbf{3} \ \mathbf{4})\}\]

The following particles form non-trivial sub fusion rings

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

Quantum Dimensions

Particle Numeric Symbolic
\(\mathbf{1}\) \(1.\) \(1\)
\(\mathbf{2}\) \(1.\) \(1\)
\(\mathbf{3}\) \(1.\) \(1\)
\(\mathbf{4}\) \(1.\) \(1\)
\(\mathbf{5}\) \(2.\) \(2\)
\(\mathcal{D}_{FP}^2\) \(8.\) \(8\)

Characters

The symbolic character table is the following

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

The numeric character table is the following

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

Modular Data

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

Adjoint Subring

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

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

Universal grading

Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{2}'\), 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

Download links for numeric data: