FR157,2\text{FR}^{7,2}_{15}

Fusion Rules

1234567231475631246754441+2+3+5+6+74+5+6+74+5+6+74+5+6+75674+5+6+71+4+5+6+72+4+5+6+73+4+5+6+76754+5+6+73+4+5+6+71+4+5+6+72+4+5+6+77564+5+6+72+4+5+6+73+4+5+6+71+4+5+6+7\begin{array}{|lllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{4} & \mathbf{7} & \mathbf{5} & \mathbf{6} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{6} & \mathbf{7} & \mathbf{5} \\ \mathbf{4} & \mathbf{4} & \mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{7} & \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \hline \end{array}

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

{(5 6 7),(5 7 6),(2 3)(5 6)}\{(\mathbf{5} \ \mathbf{6} \ \mathbf{7}), (\mathbf{5} \ \mathbf{7} \ \mathbf{6}), (\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6})\}

The following particles form non-trivial sub fusion rings

Particles SubRing
{1,2,3}\{\mathbf{1},\mathbf{2},\mathbf{3}\} Z3: FR13,2\mathbb{Z}_3:\ \text{FR}^{3,2}_{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} 3.942823.94282 Root[x33x26x+9,3]\text{Root}\left[x^3-3 x^2-6 x+9,3\right]
5\mathbf{5} 4.181944.18194 Root[x34x2x+1,3]\text{Root}\left[x^3-4 x^2-x+1,3\right]
6\mathbf{6} 4.181944.18194 Root[x34x2x+1,3]\text{Root}\left[x^3-4 x^2-x+1,3\right]
7\mathbf{7} 4.181944.18194 Root[x34x2x+1,3]\text{Root}\left[x^3-4 x^2-x+1,3\right]
DFP2\mathcal{D}_{FP}^2 71.011871.0118 Root[x33x26x+9,3]2+3Root[x34x2x+1,3]2+3\text{Root}\left[x^3-3 x^2-6 x+9,3\right]^2+3 \text{Root}\left[x^3-4 x^2-x+1,3\right]^2+3

Characters

The symbolic character table is the following

1234567111Root[x33x26x+9,3]Root[x34x2x+1,3]Root[x34x2x+1,3]Root[x34x2x+1,3]111Root[x33x26x+9,2]Root[x34x2x+1,1]Root[x34x2x+1,1]Root[x34x2x+1,1]111Root[x33x26x+9,1]Root[x34x2x+1,2]Root[x34x2x+1,2]Root[x34x2x+1,2]2110000\begin{array}{|ccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \hline 1 & 1 & 1 & \text{Root}\left[x^3-3 x^2-6 x+9,3\right] & \text{Root}\left[x^3-4 x^2-x+1,3\right] & \text{Root}\left[x^3-4 x^2-x+1,3\right] & \text{Root}\left[x^3-4 x^2-x+1,3\right] \\ 1 & 1 & 1 & \text{Root}\left[x^3-3 x^2-6 x+9,2\right] & \text{Root}\left[x^3-4 x^2-x+1,1\right] & \text{Root}\left[x^3-4 x^2-x+1,1\right] & \text{Root}\left[x^3-4 x^2-x+1,1\right] \\ 1 & 1 & 1 & \text{Root}\left[x^3-3 x^2-6 x+9,1\right] & \text{Root}\left[x^3-4 x^2-x+1,2\right] & \text{Root}\left[x^3-4 x^2-x+1,2\right] & \text{Root}\left[x^3-4 x^2-x+1,2\right] \\ 2 & -1 & -1 & 0 & 0 & 0 & 0 \\ \hline \end{array}

The numeric character table is the following

12345671.0001.0001.0003.9434.1824.1824.1821.0001.0001.0001.1110.58840.58840.58841.0001.0001.0002.0540.40640.40640.40642.0001.0001.0000000\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \hline 1.000 & 1.000 & 1.000 & 3.943 & 4.182 & 4.182 & 4.182 \\ 1.000 & 1.000 & 1.000 & 1.111 & -0.5884 & -0.5884 & -0.5884 \\ 1.000 & 1.000 & 1.000 & -2.054 & 0.4064 & 0.4064 & 0.4064 \\ 2.000 & -1.000 & -1.000 & 0 & 0 & 0 & 0 \\ \hline \end{array}

Modular Data

This fusion ring does not have any matching SS-and TT-matrices.

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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