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Z4: FR14,2\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}

Fusion Rules

1234214334214312\begin{array}{|llll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} \\ \hline \end{array}

The fusion rules are invariant under the group generated by the following permutations:

{(3 4)}\{(\mathbf{3} \ \mathbf{4})\}

The following particles form non-trivial sub fusion rings

Particles SubRing
{1,2}\{\mathbf{1},\mathbf{2}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}

Quantum 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
DFP2\mathcal{D}_{FP}^2 4.4. 44

Characters

The symbolic character table is the following

1243111111−1−11−1i−i1−1−ii\begin{array}{|cccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} \\ \hline 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & i & -i \\ 1 & -1 & -i & i \\ \hline \end{array}

The numeric character table is the following

12431.0001.0001.0001.0001.0001.000−1.000−1.0001.000−1.0001.000i−1.000i1.000−1.000−1.000i1.000i\begin{array}{|rrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & -1.000 & -1.000 \\ 1.000 & -1.000 & 1.000 i & -1.000 i \\ 1.000 & -1.000 & -1.000 i & 1.000 i \\ \hline \end{array}

Modular Data

The matching SS-matrices and twist factors are the following

SS-matrix Twist factors
12(111111−1−11−1−ii1−1i−i)\frac{1}{2}\left(\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -i & i \\ 1 & -1 & i & -i \\\end{array}\right) (0,12,18,18)(0,12,−38,−38)\begin{array}{l}\left(0,\frac{1}{2},\frac{1}{8},\frac{1}{8}\right) \\\left(0,\frac{1}{2},-\frac{3}{8},-\frac{3}{8}\right)\end{array}
12(111111−1−11−1i−i1−1−ii)\frac{1}{2}\left(\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & i & -i \\ 1 & -1 & -i & i \\\end{array}\right) (0,12,38,38)(0,12,−18,−18)\begin{array}{l}\left(0,\frac{1}{2},\frac{3}{8},\frac{3}{8}\right) \\\left(0,\frac{1}{2},-\frac{1}{8},-\frac{1}{8}\right)\end{array}

Adjoint Subring

The adjoint subring is the trivial ring.

The upper central series is the following: Z4⊃1Trivial\mathbb{Z}_4 \underset{ \mathbf{1} }{\supset} \text{Trivial}

Universal grading

Each particle can be graded as follows: deg(1)=1′,deg(2)=2′,deg(3)=3′,deg(4)=4′\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}', \text{deg}(\mathbf{4}) = \mathbf{4}', where the degrees form the group Z4\mathbb{Z}_4 with multiplication table:

1′2′3′4′2′1′4′3′3′4′2′1′4′3′1′2′\begin{array}{|llll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' \\ \mathbf{2}' & \mathbf{1}' & \mathbf{4}' & \mathbf{3}' \\ \mathbf{3}' & \mathbf{4}' & \mathbf{2}' & \mathbf{1}' \\ \mathbf{4}' & \mathbf{3}' & \mathbf{1}' & \mathbf{2}' \\ \hline \end{array}

Categorifications

Data

Download links for numeric data: