\(\text{TY}(\mathbb{Z}_3):\ \text{Potts}:\ \text{FR}^{4,2}_{2}\)
Fusion Rules
\[\begin{array}{|llll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{4} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{4} \\ \mathbf{4} & \mathbf{4} & \mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{3} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.73205\) | \(\sqrt{3}\) |
| \(\mathcal{D}_{FP}^2\) | \(6.\) | \(6\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \hline 1 & 1 & 1 & \sqrt{3} \\ 1 & 1 & 1 & -\sqrt{3} \\ 1 & e^{ 2 \pi i / 3} & e^{4 \pi i / 3 } & 0 \\ 1 & e^{4 \pi i /3 } & e^{2 \pi i / 3 } & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \hline 1.000 & 1.000 & 1.000 & 1.732 \\ 1.000 & 1.000 & 1.000 & -1.732 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 0 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 0 \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{2}, \mathbf{3}\), form the adjoint subring \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) .
The upper central series is the following: \(\text{Potts} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3} }{\supset} \mathbb{Z}_3 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{2}'\), 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: