\(\left.\text{Rep(}\text{Dic}_{12}\right):\ \text{FR}^{6,2}_{4}\)
Fusion Rules
\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} \\ \mathbf{5} & \mathbf{5} & \mathbf{6} & \mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5} \\ \mathbf{6} & \mathbf{6} & \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{6} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{3} \ \mathbf{4})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{6}\}\) | \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(2.\) | \(2\) |
| \(\mathbf{6}\) | \(2.\) | \(2\) |
| \(\mathcal{D}_{FP}^2\) | \(12.\) | \(12\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1 & 1 & 1 & 1 & 2 & 2 \\ 1 & 1 & 1 & 1 & -1 & -1 \\ 1 & 1 & -1 & -1 & -2 & 2 \\ 1 & 1 & -1 & -1 & 1 & -1 \\ 1 & -1 & i & -i & 0 & 0 \\ 1 & -1 & -i & i & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & 2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.000 & -1.000 \\ 1.000 & 1.000 & -1.000 & -1.000 & -2.000 & 2.000 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.000 & -1.000 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 0 & 0 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 0 & 0 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
This fusion ring does not provide any representations of $SL_2(\mathbb{Z}).$
Adjoint Subring
Elements \(\mathbf{1}, \mathbf{2}, \mathbf{6}\), form the adjoint subring \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) .
The upper central series is the following: \(\left.\text{Rep(}\text{Dic}_{12}\right) \underset{ \mathbf{1}, \mathbf{2}, \mathbf{6} }{\supset} \left.\text{Rep(}D_3\right)\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{2}', \text{deg}(\mathbf{6}) = \mathbf{1}'\), 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: