FR209,2\text{FR}^{9,2}_{20}

Fusion Rules

12345678921436579834126579843215678956651+4+52+3+68+97+97+865562+3+61+4+58+97+87+977778+98+91+2+3+4+7+8+95+6+7+8+95+6+7+8+989987+97+85+6+7+8+92+3+5+7+8+91+4+6+7+8+998897+87+95+6+7+8+91+4+6+7+8+92+3+5+7+8+9\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{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{5} & \mathbf{6} & \mathbf{6} & \mathbf{5} & \mathbf{1}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{6} & \mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{9} & \mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{5} & \mathbf{5} & \mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{6} & \mathbf{1}+\mathbf{4}+\mathbf{5} & \mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{9} \\ \mathbf{7} & \mathbf{7} & \mathbf{7} & \mathbf{7} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{9} & \mathbf{9} & \mathbf{8} & \mathbf{7}+\mathbf{9} & \mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{8} & \mathbf{8} & \mathbf{9} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}

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

{(2 3),(8 9)}\{(\mathbf{2} \ \mathbf{3}), (\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}
{1,3}\{\mathbf{1},\mathbf{3}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,4}\{\mathbf{1},\mathbf{4}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,4,5}\{\mathbf{1},\mathbf{4},\mathbf{5}\} 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}\} Z2×Z2: FR14,0\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}
{1,2,3,4,5,6}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\} \(\left.\mathbb{Z}_2\text{×\times Rep(}D_3\right):\ \text{FR}^{6,0}_{2}\)

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
7\mathbf{7} 4.4. 44
8\mathbf{8} 4.4. 44
9\mathbf{9} 4.4. 44
DFP2\mathcal{D}_{FP}^2 60.60. 6060

Characters

The symbolic character table is the following

1234567891111224441111221111111112111111112111111220001111000001111000001111110i3i31111110i3i3\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & 2 & 2 & 4 & 4 & 4 \\ 1 & 1 & 1 & 1 & 2 & 2 & -1 & -1 & -1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -2 & 1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1 & 2 & -1 & -1 \\ 1 & -1 & -1 & 1 & 2 & -2 & 0 & 0 & 0 \\ 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & -1 & 1 & -1 & 1 & 0 & i \sqrt{3} & -i \sqrt{3} \\ 1 & -1 & -1 & 1 & -1 & 1 & 0 & -i \sqrt{3} & i \sqrt{3} \\ \hline \end{array}

The numeric character table is the following

1234567891.0001.0001.0001.0002.0002.0004.0004.0004.0001.0001.0001.0001.0002.0002.0001.0001.0001.0001.0001.0001.0001.0001.0001.0002.0001.0001.0001.0001.0001.0001.0001.0001.0002.0001.0001.0001.0001.0001.0001.0002.0002.0000001.0001.0001.0001.000000001.0001.0001.0001.000000001.0001.0001.0001.0001.0001.00001.732i1.732i1.0001.0001.0001.0001.0001.00001.732i1.732i\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & 2.000 & 4.000 & 4.000 & 4.000 \\ 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 & -2.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.000 & -1.000 & 2.000 & -1.000 & -1.000 \\ 1.000 & -1.000 & -1.000 & 1.000 & 2.000 & -2.000 & 0 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & -1.000 & 1.000 & 0 & 1.732 i & -1.732 i \\ 1.000 & -1.000 & -1.000 & 1.000 & -1.000 & 1.000 & 0 & -1.732 i & 1.732 i \\ \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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