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Rep(Dic12): FR46,2\left.\text{Rep(}\text{Dic}_{12}\right):\ \text{FR}^{6,2}_{4}

Fusion Rules

12345621435634216543126555661+2+63+4+566553+4+51+2+6\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:

{(3 4)}\{(\mathbf{3} \ \mathbf{4})\}

The following elements form non-trivial sub fusion rings

Elements SubRing
{1,2}\{\mathbf{1},\mathbf{2}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,2,6}\{\mathbf{1},\mathbf{2},\mathbf{6}\} Rep(D3): FR23,0\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}
{1,2,3,4}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\} Z4: FR14,2\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}

Frobenius-Perron Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 1.1. 11
4\mathbf{4} 1.1. 11
5\mathbf{5} 2.2. 22
6\mathbf{6} 2.2. 22
DFP2\mathcal{D}_{FP}^2 12.12. 1212

Characters

The symbolic character table is the following

1234561111221111−1−111−1−1−2211−1−11−11−1i−i001−1−ii00\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

1234561.0001.0001.0001.0002.0002.0001.0001.0001.0001.000−1.000−1.0001.0001.000−1.000−1.000−2.0002.0001.0001.000−1.000−1.0001.000−1.0001.000−1.0001.000i−1.000i001.000−1.000−1.000i1.000i00\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 SL2(Z)SL_2(\mathbb{Z})

This fusion ring does not provide any representations of SL2(Z).SL_2(\mathbb{Z}).

Adjoint Subring

Elements 1,2,6\mathbf{1}, \mathbf{2}, \mathbf{6}, form the adjoint subring Rep(D3): FR23,0\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2} .

The upper central series is the following: Rep(Dic12)⊃1,2,6Rep(D3)\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: deg(1)=1′,deg(2)=1′,deg(3)=2′,deg(4)=2′,deg(5)=2′,deg(6)=1′\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 Z2\mathbb{Z}_2 with multiplication table:

1′2′2′1′\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}

Categorifications

Data

Download links for numeric data: