FR329,2\text{FR}^{9,2}_{32}

Fusion Rules

123456789213789456331+2+34+75+86+94+75+86+9474+71+3+4+76+8+95+8+92+3+4+75+6+95+6+8595+96+8+91+3+5+74+7+85+6+84+6+72+3+4+9686+85+8+94+7+91+3+6+75+6+92+3+4+84+5+7744+72+3+4+75+6+95+6+81+3+4+76+8+95+8+9866+85+6+92+3+4+84+5+75+8+94+7+91+3+6+7955+95+6+84+6+72+3+4+96+8+91+3+5+74+7+8\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{8} & \mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{8} & \mathbf{6}+\mathbf{9} \\ \mathbf{4} & \mathbf{7} & \mathbf{4}+\mathbf{7} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{8} \\ \mathbf{5} & \mathbf{9} & \mathbf{5}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{9} \\ \mathbf{6} & \mathbf{8} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{7} \\ \mathbf{7} & \mathbf{4} & \mathbf{4}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{6} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{7} & \mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{6}+\mathbf{7} \\ \mathbf{9} & \mathbf{5} & \mathbf{5}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}

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

{(5 6)(8 9)}\{(\mathbf{5} \ \mathbf{6}) (\mathbf{8} \ \mathbf{9})\}

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}
{1,2,3}\{\mathbf{1},\mathbf{2},\mathbf{3}\} Rep(D3): FR23,0\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}
{1,2,3,4,7}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{7}\} Rep(S4): FR65,0\left.\text{Rep(}S_4\right):\ \text{FR}^{5,0}_{6}

Frobenius-Perron Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 2.2. 22
4\mathbf{4} 3.3. 33
5\mathbf{5} 3.3. 33
6\mathbf{6} 3.3. 33
7\mathbf{7} 3.3. 33
8\mathbf{8} 3.3. 33
9\mathbf{9} 3.3. 33
DFP2\mathcal{D}_{FP}^2 60.60. 6060

Characters

The symbolic character table is the following

1234897651123333331123223221110000001101001001101111112125223434121203120\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{8} & \mathbf{9} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\ \hline 1 & 1 & 2 & 3 & 3 & 3 & 3 & 3 & 3 \\ 1 & 1 & 2 & 3 & -2 & -2 & 3 & -2 & -2 \\ 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 1 & 0 & 0 & -1 & 0 & 0 \\ 1 & -1 & 0 & -1 & 1 & 1 & 1 & -1 & -1 \\ 2 & \frac{1}{2} & \frac{5}{2} & -2 & -\frac{3}{4} & -\frac{3}{4} & -\frac{1}{2} & -\frac{1}{20} & \frac{31}{20} \\ \hline \end{array}

The numeric character table is the following

1234897651.0001.0002.0003.0003.0003.0003.0003.0003.0001.0001.0002.0003.0002.0002.0003.0002.0002.0001.0001.0001.0000000001.0001.00001.000001.000001.0001.00001.0001.0001.0001.0001.0001.0002.0000.50002.5002.0000.75000.75000.50000.050001.550\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{8} & \mathbf{9} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\ \hline 1.000 & 1.000 & 2.000 & 3.000 & 3.000 & 3.000 & 3.000 & 3.000 & 3.000 \\ 1.000 & 1.000 & 2.000 & 3.000 & -2.000 & -2.000 & 3.000 & -2.000 & -2.000 \\ 1.000 & 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & 0 & 1.000 & 0 & 0 & -1.000 & 0 & 0 \\ 1.000 & -1.000 & 0 & -1.000 & 1.000 & 1.000 & 1.000 & -1.000 & -1.000 \\ 2.000 & 0.5000 & 2.500 & -2.000 & -0.7500 & -0.7500 & -0.5000 & -0.05000 & 1.550 \\ \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

The adjoint subring is the ring itself.

The upper central series is trivial.

Universal grading

This fusion ring allows only the trivial grading.

Categorifications

Data

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