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FR107,2\text{FR}^{7,2}_{10}

Fusion Rules

123456721435673421657431265755661+2+5+73+4+6+75+6+766553+4+6+71+2+5+75+6+777775+6+75+6+71+2+3+4+5+6\begin{array}{|lllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{7} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} & \mathbf{7} \\ \mathbf{5} & \mathbf{5} & \mathbf{6} & \mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{6} & \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{7} & \mathbf{7} & \mathbf{7} & \mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \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}
{1,2,3,4}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\} Z4: FR14,2\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}

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} 2.813612.81361 Root[x3−2x2−3x+2,3]\text{Root}\left[x^3-2 x^2-3 x+2,3\right]
6\mathbf{6} 2.813612.81361 Root[x3−2x2−3x+2,3]\text{Root}\left[x^3-2 x^2-3 x+2,3\right]
7\mathbf{7} 3.102783.10278 Root[x3−2x2−6x+8,3]\text{Root}\left[x^3-2 x^2-6 x+8,3\right]
DFP2\mathcal{D}_{FP}^2 29.4629.46 2Root[x3−2x2−3x+2,3]2+Root[x3−2x2−6x+8,3]2+42 \text{Root}\left[x^3-2 x^2-3 x+2,3\right]^2+\text{Root}\left[x^3-2 x^2-6 x+8,3\right]^2+4

Characters

The symbolic character table is the following

12435671111Root[x3−2x2−3x+2,3]Root[x3−2x2−3x+2,3]Root[x3−2x2−6x+8,3]1111Root[x3−2x2−3x+2,2]Root[x3−2x2−3x+2,2]Root[x3−2x2−6x+8,1]1111Root[x3−2x2−3x+2,1]Root[x3−2x2−3x+2,1]Root[x3−2x2−6x+8,2]11−1−1−11011−1−12−201−1i−i0001−1−ii000\begin{array}{|ccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \hline 1 & 1 & 1 & 1 & \text{Root}\left[x^3-2 x^2-3 x+2,3\right] & \text{Root}\left[x^3-2 x^2-3 x+2,3\right] & \text{Root}\left[x^3-2 x^2-6 x+8,3\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-2 x^2-3 x+2,2\right] & \text{Root}\left[x^3-2 x^2-3 x+2,2\right] & \text{Root}\left[x^3-2 x^2-6 x+8,1\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-2 x^2-3 x+2,1\right] & \text{Root}\left[x^3-2 x^2-3 x+2,1\right] & \text{Root}\left[x^3-2 x^2-6 x+8,2\right] \\ 1 & 1 & -1 & -1 & -1 & 1 & 0 \\ 1 & 1 & -1 & -1 & 2 & -2 & 0 \\ 1 & -1 & i & -i & 0 & 0 & 0 \\ 1 & -1 & -i & i & 0 & 0 & 0 \\ \hline \end{array}

The numeric character table is the following

12435671.0001.0001.0001.0002.8142.8143.1031.0001.0001.0001.0000.52930.5293−2.2491.0001.0001.0001.000−1.343−1.3431.1461.0001.000−1.000−1.000−1.0001.00001.0001.000−1.000−1.0002.000−2.00001.000−1.0001.000i−1.000i0001.000−1.000−1.000i1.000i000\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.814 & 2.814 & 3.103 \\ 1.000 & 1.000 & 1.000 & 1.000 & 0.5293 & 0.5293 & -2.249 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.343 & -1.343 & 1.146 \\ 1.000 & 1.000 & -1.000 & -1.000 & -1.000 & 1.000 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 2.000 & -2.000 & 0 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 0 & 0 & 0 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 0 & 0 & 0 \\ \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

This fusion ring has no categorifications because of the extended cyclotomic criterion.

Data

Download links for numeric data: