AnyonWiki

FR299,0\text{FR}^{9,0}_{29}

Fusion Rules

12345678921437856934128765943216587957861+6+7+8+94+5+7+8+92+5+6+8+93+5+6+7+95+6+7+8+968754+5+7+8+91+6+7+8+93+5+6+7+92+5+6+8+95+6+7+8+975682+5+6+8+93+5+6+7+91+6+7+8+94+5+7+8+95+6+7+8+986573+5+6+7+92+5+6+8+94+5+7+8+91+6+7+8+95+6+7+8+999995+6+7+8+95+6+7+8+95+6+7+8+95+6+7+8+91+2+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{4} & \mathbf{3} & \mathbf{7} & \mathbf{8} & \mathbf{5} & \mathbf{6} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{5} & \mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{5}+\mathbf{6}+\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{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\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:

{(2 3)(7 8),(2 4)(6 7),(3 4)(6 8)}\{(\mathbf{2} \ \mathbf{3}) (\mathbf{7} \ \mathbf{8}), (\mathbf{2} \ \mathbf{4}) (\mathbf{6} \ \mathbf{7}), (\mathbf{3} \ \mathbf{4}) (\mathbf{6} \ \mathbf{8})\}

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}
{1,2,3,4}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\} Z2×Z2: FR14,0\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}

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} 4.402684.40268 Root[x3−4x2−2x+1,3]\text{Root}\left[x^3-4 x^2-2 x+1,3\right]
6\mathbf{6} 4.402684.40268 Root[x3−4x2−2x+1,3]\text{Root}\left[x^3-4 x^2-2 x+1,3\right]
7\mathbf{7} 4.402684.40268 Root[x3−4x2−2x+1,3]\text{Root}\left[x^3-4 x^2-2 x+1,3\right]
8\mathbf{8} 4.402684.40268 Root[x3−4x2−2x+1,3]\text{Root}\left[x^3-4 x^2-2 x+1,3\right]
9\mathbf{9} 5.175545.17554 Root[x3−5x2−4x+16,3]\text{Root}\left[x^3-5 x^2-4 x+16,3\right]
DFP2\mathcal{D}_{FP}^2 108.321108.321 4Root[x3−4x2−2x+1,3]2+Root[x3−5x2−4x+16,3]2+44 \text{Root}\left[x^3-4 x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-5 x^2-4 x+16,3\right]^2+4

Characters

The symbolic character table is the following

1243675891111Root[x3−4x2−2x+1,3]Root[x3−4x2−2x+1,3]Root[x3−4x2−2x+1,3]Root[x3−4x2−2x+1,3]Root[x3−5x2−4x+16,3]1111Root[x3−4x2−2x+1,2]Root[x3−4x2−2x+1,2]Root[x3−4x2−2x+1,2]Root[x3−4x2−2x+1,2]Root[x3−5x2−4x+16,1]1111Root[x3−4x2−2x+1,1]Root[x3−4x2−2x+1,1]Root[x3−4x2−2x+1,1]Root[x3−4x2−2x+1,1]Root[x3−5x2−4x+16,2]1−11−112(−1−5)12(1+5)12(−1−5)12(1+5)011−1−112(1−5)12(5−1)12(5−1)12(1−5)01−11−112(5−1)12(1−5)12(5−1)12(1−5)01−1−1112(1+5)12(1+5)12(−1−5)12(−1−5)011−1−112(1+5)12(−1−5)12(−1−5)12(1+5)01−1−1112(1−5)12(1−5)12(5−1)12(5−1)0\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-2 x+1,3\right] & \text{Root}\left[x^3-4 x^2-2 x+1,3\right] & \text{Root}\left[x^3-4 x^2-2 x+1,3\right] & \text{Root}\left[x^3-4 x^2-2 x+1,3\right] & \text{Root}\left[x^3-5 x^2-4 x+16,3\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-2 x+1,2\right] & \text{Root}\left[x^3-4 x^2-2 x+1,2\right] & \text{Root}\left[x^3-4 x^2-2 x+1,2\right] & \text{Root}\left[x^3-4 x^2-2 x+1,2\right] & \text{Root}\left[x^3-5 x^2-4 x+16,1\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-2 x+1,1\right] & \text{Root}\left[x^3-4 x^2-2 x+1,1\right] & \text{Root}\left[x^3-4 x^2-2 x+1,1\right] & \text{Root}\left[x^3-4 x^2-2 x+1,1\right] & \text{Root}\left[x^3-5 x^2-4 x+16,2\right] \\ 1 & -1 & 1 & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 0 \\ 1 & 1 & -1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 0 \\ 1 & -1 & 1 & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 0 \\ 1 & -1 & -1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 0 \\ 1 & 1 & -1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 0 \\ 1 & -1 & -1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 0 \\ \hline \end{array}

The numeric character table is the following

1243675891.0001.0001.0001.0004.4034.4034.4034.4035.1761.0001.0001.0001.0000.31600.31600.31600.3160−1.8481.0001.0001.0001.000−0.7187−0.7187−0.7187−0.71871.6731.000−1.0001.000−1.000−1.6181.618−1.6181.61801.0001.000−1.000−1.000−0.61800.61800.6180−0.618001.000−1.0001.000−1.0000.6180−0.61800.6180−0.618001.000−1.000−1.0001.0001.6181.618−1.618−1.61801.0001.000−1.000−1.0001.618−1.618−1.6181.61801.000−1.000−1.0001.000−0.6180−0.61800.61800.61800\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 4.403 & 4.403 & 4.403 & 4.403 & 5.176 \\ 1.000 & 1.000 & 1.000 & 1.000 & 0.3160 & 0.3160 & 0.3160 & 0.3160 & -1.848 \\ 1.000 & 1.000 & 1.000 & 1.000 & -0.7187 & -0.7187 & -0.7187 & -0.7187 & 1.673 \\ 1.000 & -1.000 & 1.000 & -1.000 & -1.618 & 1.618 & -1.618 & 1.618 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & -0.6180 & 0.6180 & 0.6180 & -0.6180 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0.6180 & -0.6180 & 0.6180 & -0.6180 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 1.618 & 1.618 & -1.618 & -1.618 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.618 & -1.618 & -1.618 & 1.618 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & -0.6180 & -0.6180 & 0.6180 & 0.6180 & 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: