\(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\)
Fusion Rules
\[\begin{array}{|lll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3})\}\]Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathcal{D}_{FP}^2\) | \(3.\) | \(3\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline 1 & 1 & 1 \\ 1 & e^{2 \pi i /3 } & e^{4 \pi i /3 } \\ 1 & e^{4 \pi i /3 } & e^{ 2 \pi i / 3 } \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline 1.000 & 1.000 & 1.000 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{3}}\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & - e^{ i \pi / 3 } & e^{2\pi i /3 } \\ 1 & e^{2 \pi i/3} & - e^{ \pi i / 3 } \\\end{array}\right)\) | \(\begin{array}{l}\left(0,-\frac{1}{3},-\frac{1}{3}\right)\end{array}\) |
| \(\frac{1}{\sqrt{3}}\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & e^{2\pi i /3 } & - e^{ \pi i/ 3 } \\ 1 & - e^{ \pi i/ 3 } & e^{2 \pi i / 3} \\\end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{3},\frac{1}{3}\right)\end{array}\) |
Adjoint Subring
The adjoint subring is the trivial ring.
The upper central series is the following: \(\mathbb{Z}_3 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}'\), where the degrees form the group \(\mathbb{Z}_3\) with multiplication table:
\[\begin{array}{|lll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' \\ \mathbf{2}' & \mathbf{3}' & \mathbf{1}' \\ \mathbf{3}' & \mathbf{1}' & \mathbf{2}' \\ \hline \end{array}\]Categorifications
Data
Download links for numeric data: