AnyonWiki

FR389,0\text{FR}^{9,0}_{38}

Fusion Rules

123456789213456789331+2+43+48+97+96+85+75+6443+41+2+36+75+85+96+97+8558+96+71+2+6+7+8+94+5+6+8+94+5+7+8+93+5+6+7+83+5+6+7+9667+95+84+5+6+8+91+2+5+6+7+83+6+7+8+94+5+6+7+93+5+7+8+9776+85+94+5+7+8+93+6+7+8+91+2+5+6+7+93+5+6+8+94+5+6+7+8885+76+93+5+6+7+84+5+6+7+93+5+6+8+91+2+5+7+8+94+6+7+8+9995+67+83+5+6+7+93+5+7+8+94+5+6+7+84+6+7+8+91+2+5+6+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{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{4} & \mathbf{3}+\mathbf{4} & \mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{7} & \mathbf{5}+\mathbf{6} \\ \mathbf{4} & \mathbf{4} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{8} & \mathbf{5}+\mathbf{9} & \mathbf{6}+\mathbf{9} & \mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{5} & \mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} \\ \mathbf{6} & \mathbf{6} & \mathbf{7}+\mathbf{9} & \mathbf{5}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{8} & \mathbf{5}+\mathbf{7} & \mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{5}+\mathbf{6} & \mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}

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

{(3 4)(6 8 7 9),(3 4)(6 9 7 8)}\{(\mathbf{3} \ \mathbf{4}) (\mathbf{6} \ \mathbf{8} \ \mathbf{7} \ \mathbf{9}), (\mathbf{3} \ \mathbf{4}) (\mathbf{6} \ \mathbf{9} \ \mathbf{7} \ \mathbf{8})\}

The following particles form non-trivial sub fusion rings

Particles 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}\} Rep(D5): FR34,0\left.\text{Rep(}D_5\right):\ \text{FR}^{4,0}_{3}

Quantum Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 2.2. 22
4\mathbf{4} 2.2. 22
5\mathbf{5} 4.449494.44949 2+62+\sqrt{6}
6\mathbf{6} 4.449494.44949 2+62+\sqrt{6}
7\mathbf{7} 4.449494.44949 2+62+\sqrt{6}
8\mathbf{8} 4.449494.44949 2+62+\sqrt{6}
9\mathbf{9} 4.449494.44949 2+62+\sqrt{6}
DFP2\mathcal{D}_{FP}^2 108.99108.99 10+5(2+6)210+5 \left(2+\sqrt{6}\right)^2

Characters

The symbolic character table is the following

12345678911222+62+62+62+62+611222−62−62−62−62−61112(5−1)12(−1−5)−212(1+5)12(1+5)12(1−5)12(1−5)1112(5−1)12(−1−5)1Root[x4+x3−4x2−4x+1,1]Root[x4+x3−4x2−4x+1,3]Root[x4+x3−4x2−4x+1,4]Root[x4+x3−4x2−4x+1,2]1112(−1−5)12(5−1)−212(1−5)12(1−5)12(1+5)12(1+5)1112(−1−5)12(5−1)1Root[x4+x3−4x2−4x+1,2]Root[x4+x3−4x2−4x+1,4]Root[x4+x3−4x2−4x+1,1]Root[x4+x3−4x2−4x+1,3]1112(5−1)12(−1−5)1Root[x4+x3−4x2−4x+1,3]Root[x4+x3−4x2−4x+1,1]Root[x4+x3−4x2−4x+1,2]Root[x4+x3−4x2−4x+1,4]1112(−1−5)12(5−1)1Root[x4+x3−4x2−4x+1,4]Root[x4+x3−4x2−4x+1,2]Root[x4+x3−4x2−4x+1,3]Root[x4+x3−4x2−4x+1,1]1−10000000\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 & 2 & 2 & 2+\sqrt{6} & 2+\sqrt{6} & 2+\sqrt{6} & 2+\sqrt{6} & 2+\sqrt{6} \\ 1 & 1 & 2 & 2 & 2-\sqrt{6} & 2-\sqrt{6} & 2-\sqrt{6} & 2-\sqrt{6} & 2-\sqrt{6} \\ 1 & 1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & -2 & \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) \\ 1 & 1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 1 & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] \\ 1 & 1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & -2 & \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) \\ 1 & 1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 1 & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] \\ 1 & 1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 1 & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] \\ 1 & 1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 1 & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}

The numeric character table is the following

1234567891.0001.0002.0002.0004.4494.4494.4494.4494.4491.0001.0002.0002.000−0.4495−0.4495−0.4495−0.4495−0.44951.0001.0000.6180−1.618−2.0001.6181.618−0.6180−0.61801.0001.0000.6180−1.6181.000−1.8270.20911.956−1.3381.0001.000−1.6180.6180−2.000−0.6180−0.61801.6181.6181.0001.000−1.6180.61801.000−1.3381.956−1.8270.20911.0001.0000.6180−1.6181.0000.2091−1.827−1.3381.9561.0001.000−1.6180.61801.0001.956−1.3380.2091−1.8271.000−1.0000000000\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 & 2.000 & 2.000 & 4.449 & 4.449 & 4.449 & 4.449 & 4.449 \\ 1.000 & 1.000 & 2.000 & 2.000 & -0.4495 & -0.4495 & -0.4495 & -0.4495 & -0.4495 \\ 1.000 & 1.000 & 0.6180 & -1.618 & -2.000 & 1.618 & 1.618 & -0.6180 & -0.6180 \\ 1.000 & 1.000 & 0.6180 & -1.618 & 1.000 & -1.827 & 0.2091 & 1.956 & -1.338 \\ 1.000 & 1.000 & -1.618 & 0.6180 & -2.000 & -0.6180 & -0.6180 & 1.618 & 1.618 \\ 1.000 & 1.000 & -1.618 & 0.6180 & 1.000 & -1.338 & 1.956 & -1.827 & 0.2091 \\ 1.000 & 1.000 & 0.6180 & -1.618 & 1.000 & 0.2091 & -1.827 & -1.338 & 1.956 \\ 1.000 & 1.000 & -1.618 & 0.6180 & 1.000 & 1.956 & -1.338 & 0.2091 & -1.827 \\ 1.000 & -1.000 & 0 & 0 & 0 & 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

This fusion ring has no pivotal categorifications because of the pivotal version of the Drinfeld center criterion.

Data

Download links for numeric data: