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FR239,0\text{FR}^{9,0}_{23}

Fusion Rules

12345678921435768934125769843215679855551+2+3+46+76+78+98+967766+71+4+5+8+92+3+5+8+96+7+8+96+7+8+976676+72+3+5+8+91+4+5+8+96+7+8+96+7+8+988998+96+7+8+96+7+8+91+2+5+6+7+8+93+4+5+6+7+8+999888+96+7+8+96+7+8+93+4+5+6+7+8+91+2+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{5} & \mathbf{7} & \mathbf{6} & \mathbf{8} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{8} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{7} & \mathbf{7} & \mathbf{6} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{6} & \mathbf{6} & \mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{8} & \mathbf{9} & \mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{8} & \mathbf{8} & \mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\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:

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

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}
{1,2,3,4,5}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5}\} Rep(D4): FR15,0\left.\text{Rep(}D_4\right):\ \text{FR}^{5,0}_{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} 2.2. 22
6\mathbf{6} 3.603883.60388 Root[x3−2x2−8x+8,3]\text{Root}\left[x^3-2 x^2-8 x+8,3\right]
7\mathbf{7} 3.603883.60388 Root[x3−2x2−8x+8,3]\text{Root}\left[x^3-2 x^2-8 x+8,3\right]
8\mathbf{8} 4.493964.49396 Root[x3−4x2−4x+8,3]\text{Root}\left[x^3-4 x^2-4 x+8,3\right]
9\mathbf{9} 4.493964.49396 Root[x3−4x2−4x+8,3]\text{Root}\left[x^3-4 x^2-4 x+8,3\right]
DFP2\mathcal{D}_{FP}^2 74.367274.3672 2Root[x3−2x2−8x+8,3]2+2Root[x3−4x2−4x+8,3]2+82 \text{Root}\left[x^3-2 x^2-8 x+8,3\right]^2+2 \text{Root}\left[x^3-4 x^2-4 x+8,3\right]^2+8

Characters

The symbolic character table is the following

13425678911112Root[x3−2x2−8x+8,3]Root[x3−2x2−8x+8,3]Root[x3−4x2−4x+8,3]Root[x3−4x2−4x+8,3]11112Root[x3−2x2−8x+8,2]Root[x3−2x2−8x+8,2]Root[x3−4x2−4x+8,1]Root[x3−4x2−4x+8,1]11112Root[x3−2x2−8x+8,1]Root[x3−2x2−8x+8,1]Root[x3−4x2−4x+8,2]Root[x3−4x2−4x+8,2]1111−200001−11−10−22001−11−102−20011−1−1000001−1−110002−21−1−11000−22\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & 2 & \text{Root}\left[x^3-2 x^2-8 x+8,3\right] & \text{Root}\left[x^3-2 x^2-8 x+8,3\right] & \text{Root}\left[x^3-4 x^2-4 x+8,3\right] & \text{Root}\left[x^3-4 x^2-4 x+8,3\right] \\ 1 & 1 & 1 & 1 & 2 & \text{Root}\left[x^3-2 x^2-8 x+8,2\right] & \text{Root}\left[x^3-2 x^2-8 x+8,2\right] & \text{Root}\left[x^3-4 x^2-4 x+8,1\right] & \text{Root}\left[x^3-4 x^2-4 x+8,1\right] \\ 1 & 1 & 1 & 1 & 2 & \text{Root}\left[x^3-2 x^2-8 x+8,1\right] & \text{Root}\left[x^3-2 x^2-8 x+8,1\right] & \text{Root}\left[x^3-4 x^2-4 x+8,2\right] & \text{Root}\left[x^3-4 x^2-4 x+8,2\right] \\ 1 & 1 & 1 & 1 & -2 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & 0 & -\sqrt{2} & \sqrt{2} & 0 & 0 \\ 1 & -1 & 1 & -1 & 0 & \sqrt{2} & -\sqrt{2} & 0 & 0 \\ 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & -1 & 1 & 0 & 0 & 0 & \sqrt{2} & -\sqrt{2} \\ 1 & -1 & -1 & 1 & 0 & 0 & 0 & -\sqrt{2} & \sqrt{2} \\ \hline \end{array}

The numeric character table is the following

1342567891.0001.0001.0001.0002.0003.6043.6044.4944.4941.0001.0001.0001.0002.0000.89010.8901−1.604−1.6041.0001.0001.0001.0002.000−2.494−2.4941.1101.1101.0001.0001.0001.000−2.00000001.000−1.0001.000−1.0000−1.4141.414001.000−1.0001.000−1.00001.414−1.414001.0001.000−1.000−1.000000001.000−1.000−1.0001.0000001.414−1.4141.000−1.000−1.0001.000000−1.4141.414\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & 3.604 & 3.604 & 4.494 & 4.494 \\ 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & 0.8901 & 0.8901 & -1.604 & -1.604 \\ 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & -2.494 & -2.494 & 1.110 & 1.110 \\ 1.000 & 1.000 & 1.000 & 1.000 & -2.000 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 & -1.414 & 1.414 & 0 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 & 1.414 & -1.414 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 & 0 & 1.414 & -1.414 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 & 0 & -1.414 & 1.414 \\ \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

Data

Download links for numeric data: