SU(2)4: FR35,0\text{SU(2})_4:\ \text{FR}^{5,0}_{3}

Fusion Rules

1234521435341+52+53+4432+51+53+4553+43+41+2+5\begin{array}{|lllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{5} & \mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{5} & \mathbf{1}+\mathbf{5} & \mathbf{3}+\mathbf{4} \\ \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{5} \\ \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}
{1,2,5}\{\mathbf{1},\mathbf{2},\mathbf{5}\} Rep(D3): FR23,0\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}

Frobenius-Perron Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 1.732051.73205 3\sqrt{3}
4\mathbf{4} 1.732051.73205 3\sqrt{3}
5\mathbf{5} 2.2. 22
DFP2\mathcal{D}_{FP}^2 12.12. 1212

Characters

The symbolic character table is the following

123451133211001113321111011110\begin{array}{|ccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & 0 & 0 & -1 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ 1 & -1 & 1 & -1 & 0 \\ 1 & -1 & -1 & 1 & 0 \\ \hline \end{array}

The numeric character table is the following

123451.0001.0001.7321.7322.0001.0001.000001.0001.0001.0001.7321.7322.0001.0001.0001.0001.00001.0001.0001.0001.0000\begin{array}{|rrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline 1.000 & 1.000 & 1.732 & 1.732 & 2.000 \\ 1.000 & 1.000 & 0 & 0 & -1.000 \\ 1.000 & 1.000 & -1.732 & -1.732 & 2.000 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 \\ \hline \end{array}

Representations of SL2(Z)SL_2(\mathbb{Z})

The matching SS-matrices and twist factors are the following

SS-matrix Twist factors
123(1133211332333303333022002)\frac{1}{2 \sqrt{3}}\left(\begin{array}{ccccc} 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ \sqrt{3} & -\sqrt{3} & \sqrt{3} & -\sqrt{3} & 0 \\ \sqrt{3} & -\sqrt{3} & -\sqrt{3} & \sqrt{3} & 0 \\ 2 & 2 & 0 & 0 & -2 \\\end{array}\right) (0,0,38,18,13)(0,0,18,38,13)(0,0,18,38,13)(0,0,38,18,13)\begin{array}{l}\left(0,0,\frac{3}{8},-\frac{1}{8},-\frac{1}{3}\right) \\\left(0,0,-\frac{1}{8},\frac{3}{8},-\frac{1}{3}\right) \\\left(0,0,\frac{1}{8},-\frac{3}{8},\frac{1}{3}\right) \\\left(0,0,-\frac{3}{8},\frac{1}{8},\frac{1}{3}\right)\end{array}
123(1133211332333303333022002)\frac{1}{2 \sqrt{3}}\left(\begin{array}{ccccc} 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ \sqrt{3} & -\sqrt{3} & -\sqrt{3} & \sqrt{3} & 0 \\ \sqrt{3} & -\sqrt{3} & \sqrt{3} & -\sqrt{3} & 0 \\ 2 & 2 & 0 & 0 & -2 \\\end{array}\right) (0,0,18,38,13)(0,0,38,18,13)(0,0,38,18,13)(0,0,18,38,13)\begin{array}{l}\left(0,0,-\frac{1}{8},\frac{3}{8},\frac{1}{3}\right) \\\left(0,0,\frac{3}{8},-\frac{1}{8},\frac{1}{3}\right) \\\left(0,0,-\frac{3}{8},\frac{1}{8},-\frac{1}{3}\right) \\\left(0,0,\frac{1}{8},-\frac{3}{8},-\frac{1}{3}\right)\end{array}

Adjoint Subring

Elements 1,2,5\mathbf{1}, \mathbf{2}, \mathbf{5}, 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: SU(2)41,2,5Rep(D3)\text{SU(2})_4 \underset{ \mathbf{1}, \mathbf{2}, \mathbf{5} }{\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)=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{1}', where the degrees form the group Z2\mathbb{Z}_2 with multiplication table:

1221\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}

Categorifications

Data

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