\(\mathbb{Z}_5:\ \text{FR}^{5,4}_{1}\)
Fusion Rules
\[\begin{array}{|lllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \mathbf{2} & \mathbf{5} & \mathbf{1} & \mathbf{3} & \mathbf{4} \\ \mathbf{3} & \mathbf{1} & \mathbf{4} & \mathbf{5} & \mathbf{2} \\ \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{2} & \mathbf{1} \\ \mathbf{5} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{3} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{4} \ \mathbf{3} \ \mathbf{5}), (\mathbf{2} \ \mathbf{5} \ \mathbf{3} \ \mathbf{4})\}\]Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(1.\) | \(1\) |
| \(\mathcal{D}_{FP}^2\) | \(5.\) | \(5\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccc|} \hline \mathbf{1} & \mathbf{3} & \mathbf{2} & \mathbf{5} & \mathbf{4} \\ \hline 1 & 1 & 1 & 1 & 1 \\ 1 & e^{-\frac{2 i \pi }{5}} & e^{\frac{2 i \pi }{5}} & e^{\frac{4 i \pi }{5}} & e^{-\frac{4 i \pi }{5}} \\ 1 & e^{\frac{2 i \pi }{5}} & e^{-\frac{2 i \pi }{5}} & e^{-\frac{4 i \pi }{5}} & e^{\frac{4 i \pi }{5}} \\ 1 & e^{\frac{4 i \pi }{5}} & e^{-\frac{4 i \pi }{5}} & e^{\frac{2 i \pi }{5}} & e^{-\frac{2 i \pi }{5}} \\ 1 & e^{-\frac{4 i \pi }{5}} & e^{\frac{4 i \pi }{5}} & e^{-\frac{2 i \pi }{5}} & e^{\frac{2 i \pi }{5}} \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrr|} \hline \mathbf{1} & \mathbf{3} & \mathbf{2} & \mathbf{5} & \mathbf{4} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 1.000 & 0.3090-0.9511 i & 0.3090+0.9511 i & -0.8090+0.5878 i & -0.8090-0.5878 i \\ 1.000 & 0.3090+0.9511 i & 0.3090-0.9511 i & -0.8090-0.5878 i & -0.8090+0.5878 i \\ 1.000 & -0.8090+0.5878 i & -0.8090-0.5878 i & 0.3090+0.9511 i & 0.3090-0.9511 i \\ 1.000 & -0.8090-0.5878 i & -0.8090+0.5878 i & 0.3090-0.9511 i & 0.3090+0.9511 i \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{5}}\left(\begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 1 & e^{\frac{4 i \pi }{5}} & e^{-\frac{4 i \pi }{5}} & e^{\frac{2 i \pi }{5}} & e^{-\frac{2 i \pi }{5}} \\ 1 & e^{-\frac{4 i \pi }{5}} & e^{\frac{4 i \pi }{5}} & e^{-\frac{2 i \pi }{5}} & e^{\frac{2 i \pi }{5}} \\ 1 & e^{\frac{2 i \pi }{5}} & e^{-\frac{2 i \pi }{5}} & e^{-\frac{4 i \pi }{5}} & e^{\frac{4 i \pi }{5}} \\ 1 & e^{-\frac{2 i \pi }{5}} & e^{\frac{2 i \pi }{5}} & e^{\frac{4 i \pi }{5}} & e^{-\frac{4 i \pi }{5}} \\\end{array}\right)\) | \(\begin{array}{l}\left(0,-\frac{1}{5},-\frac{1}{5},\frac{1}{5},\frac{1}{5}\right)\end{array}\) |
| \(\frac{1}{\sqrt{5}}\left(\begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 1 & e^{\frac{2 i \pi }{5}} & e^{-\frac{2 i \pi }{5}} & e^{-\frac{4 i \pi }{5}} & e^{\frac{4 i \pi }{5}} \\ 1 & e^{-\frac{2 i \pi }{5}} & e^{\frac{2 i \pi }{5}} & e^{\frac{4 i \pi }{5}} & e^{-\frac{4 i \pi }{5}} \\ 1 & e^{-\frac{4 i \pi }{5}} & e^{\frac{4 i \pi }{5}} & e^{-\frac{2 i \pi }{5}} & e^{\frac{2 i \pi }{5}} \\ 1 & e^{\frac{4 i \pi }{5}} & e^{-\frac{4 i \pi }{5}} & e^{\frac{2 i \pi }{5}} & e^{-\frac{2 i \pi }{5}} \\\end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{2}{5},\frac{2}{5},-\frac{2}{5},-\frac{2}{5}\right)\end{array}\) |
Adjoint Subring
The adjoint subring is the trivial ring.
The upper central series is the following: \(\mathbb{Z}_5 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}', \text{deg}(\mathbf{4}) = \mathbf{4}', \text{deg}(\mathbf{5}) = \mathbf{5}'\), where the degrees form the group \(\mathbb{Z}_5\) with multiplication table:
\[\begin{array}{|lllll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' & \mathbf{5}' \\ \mathbf{2}' & \mathbf{5}' & \mathbf{1}' & \mathbf{3}' & \mathbf{4}' \\ \mathbf{3}' & \mathbf{1}' & \mathbf{4}' & \mathbf{5}' & \mathbf{2}' \\ \mathbf{4}' & \mathbf{3}' & \mathbf{5}' & \mathbf{2}' & \mathbf{1}' \\ \mathbf{5}' & \mathbf{4}' & \mathbf{2}' & \mathbf{1}' & \mathbf{3}' \\ \hline \end{array}\]Categorifications
Data
Download links for numeric data: