AnyonWiki

FR459,0\text{FR}^{9,0}_{45}

Fusion Rules

123456789217654398371+4+5+6+7+93+4+7+8+93+5+7+8+93+6+7+8+92+3+4+5+6+84+5+6+7+8+93+4+5+6+8+9463+4+7+8+91+3+4+5+6+94+5+6+8+92+4+5+6+7+83+6+7+8+93+5+6+7+8+93+4+5+7+8+9553+5+7+8+94+5+6+8+91+2+3+4+5+6+74+5+6+8+93+5+7+8+93+4+6+7+8+93+4+6+7+8+9643+6+7+8+92+4+5+6+7+84+5+6+8+91+3+4+5+6+93+4+7+8+93+4+5+7+8+93+5+6+7+8+9732+3+4+5+6+83+6+7+8+93+5+7+8+93+4+7+8+91+4+5+6+7+93+4+5+6+8+94+5+6+7+8+9894+5+6+7+8+93+5+6+7+8+93+4+6+7+8+93+4+5+7+8+93+4+5+6+8+91+3+4+5+6+7+8+92+3+4+5+6+7+8+9983+4+5+6+8+93+4+5+7+8+93+4+6+7+8+93+5+6+7+8+94+5+6+7+8+92+3+4+5+6+7+8+91+3+4+5+6+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{7} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{9} & \mathbf{8} \\ \mathbf{3} & \mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} \\ \mathbf{4} & \mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{4} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{3} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}

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

{(4 6)}\{(\mathbf{4} \ \mathbf{6})\}

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}

Frobenius-Perron Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 5.372285.37228 12(5+33)\frac{1}{2} \left(5+\sqrt{33}\right)
4\mathbf{4} 5.372285.37228 12(5+33)\frac{1}{2} \left(5+\sqrt{33}\right)
5\mathbf{5} 5.372285.37228 12(5+33)\frac{1}{2} \left(5+\sqrt{33}\right)
6\mathbf{6} 5.372285.37228 12(5+33)\frac{1}{2} \left(5+\sqrt{33}\right)
7\mathbf{7} 5.372285.37228 12(5+33)\frac{1}{2} \left(5+\sqrt{33}\right)
8\mathbf{8} 6.372286.37228 12(7+33)\frac{1}{2} \left(7+\sqrt{33}\right)
9\mathbf{9} 6.372286.37228 12(7+33)\frac{1}{2} \left(7+\sqrt{33}\right)
DFP2\mathcal{D}_{FP}^2 227.519227.519 2+54(5+33)2+12(7+33)22+\frac{5}{4} \left(5+\sqrt{33}\right)^2+\frac{1}{2} \left(7+\sqrt{33}\right)^2

Characters

The symbolic character table is the following

1234567981112(5+33)12(5+33)12(5+33)12(5+33)12(5+33)12(7+33)12(7+33)1111−211−1−111−2222−2−1−1111−12−11−1−11112(5−33)12(5−33)12(5−33)12(5−33)12(5−33)12(7−33)12(7−33)1−1−200021−11−1130−3−11−11−11−303−11−11−100000−11\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1 & 1 & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(7+\sqrt{33}\right) & \frac{1}{2} \left(7+\sqrt{33}\right) \\ 1 & 1 & 1 & 1 & -2 & 1 & 1 & -1 & -1 \\ 1 & 1 & -2 & 2 & 2 & 2 & -2 & -1 & -1 \\ 1 & 1 & 1 & -1 & 2 & -1 & 1 & -1 & -1 \\ 1 & 1 & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(7-\sqrt{33}\right) & \frac{1}{2} \left(7-\sqrt{33}\right) \\ 1 & -1 & -2 & 0 & 0 & 0 & 2 & 1 & -1 \\ 1 & -1 & 1 & \sqrt{3} & 0 & -\sqrt{3} & -1 & 1 & -1 \\ 1 & -1 & 1 & -\sqrt{3} & 0 & \sqrt{3} & -1 & 1 & -1 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 1 \\ \hline \end{array}

The numeric character table is the following

1234567981.0001.0005.3725.3725.3725.3725.3726.3726.3721.0001.0001.0001.000−2.0001.0001.000−1.000−1.0001.0001.000−2.0002.0002.0002.000−2.000−1.000−1.0001.0001.0001.000−1.0002.000−1.0001.000−1.000−1.0001.0001.000−0.3723−0.3723−0.3723−0.3723−0.37230.62770.62771.000−1.000−2.0000002.0001.000−1.0001.000−1.0001.0001.7320−1.732−1.0001.000−1.0001.000−1.0001.000−1.73201.732−1.0001.000−1.0001.000−1.00000000−1.0001.000\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1.000 & 1.000 & 5.372 & 5.372 & 5.372 & 5.372 & 5.372 & 6.372 & 6.372 \\ 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 & 2.000 & 2.000 & -2.000 & -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 & -0.3723 & -0.3723 & -0.3723 & -0.3723 & -0.3723 & 0.6277 & 0.6277 \\ 1.000 & -1.000 & -2.000 & 0 & 0 & 0 & 2.000 & 1.000 & -1.000 \\ 1.000 & -1.000 & 1.000 & 1.732 & 0 & -1.732 & -1.000 & 1.000 & -1.000 \\ 1.000 & -1.000 & 1.000 & -1.732 & 0 & 1.732 & -1.000 & 1.000 & -1.000 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 & -1.000 & 1.000 \\ \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 dd-number criterion.

Data

Download links for numeric data: