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FR55,0\text{FR}^{5,0}_{5}

Fusion Rules

1234521354331+2+34+54+5454+51+3+52+3+4544+52+3+41+3+5\begin{array}{|lllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{5} & \mathbf{4} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{4}+\mathbf{5} & \mathbf{4}+\mathbf{5} \\ \mathbf{4} & \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4} \\ \mathbf{5} & \mathbf{4} & \mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{3}+\mathbf{5} \\ \hline \end{array}

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}\{\mathbf{1},\mathbf{2},\mathbf{3}\} Rep(D3): FR23,0\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}

Frobenius-Perron 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} 2.302782.30278 12(1+13)\frac{1}{2} \left(1+\sqrt{13}\right)
5\mathbf{5} 2.302782.30278 12(1+13)\frac{1}{2} \left(1+\sqrt{13}\right)
DFP2\mathcal{D}_{FP}^2 16.605616.6056 6+12(1+13)26+\frac{1}{2} \left(1+\sqrt{13}\right)^2

Characters

The symbolic character table is the following

1235411212(1+13)12(1+13)11212(1−13)12(1−13)11−1001−1012(1+5)12(−1−5)1−1012(1−5)12(5−1)\begin{array}{|ccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{4} \\ \hline 1 & 1 & 2 & \frac{1}{2} \left(1+\sqrt{13}\right) & \frac{1}{2} \left(1+\sqrt{13}\right) \\ 1 & 1 & 2 & \frac{1}{2} \left(1-\sqrt{13}\right) & \frac{1}{2} \left(1-\sqrt{13}\right) \\ 1 & 1 & -1 & 0 & 0 \\ 1 & -1 & 0 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & -1 & 0 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) \\ \hline \end{array}

The numeric character table is the following

123541.0001.0002.0002.3032.3031.0001.0002.000−1.303−1.3031.0001.000−1.000001.000−1.00001.618−1.6181.000−1.0000−0.61800.6180\begin{array}{|rrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{4} \\ \hline 1.000 & 1.000 & 2.000 & 2.303 & 2.303 \\ 1.000 & 1.000 & 2.000 & -1.303 & -1.303 \\ 1.000 & 1.000 & -1.000 & 0 & 0 \\ 1.000 & -1.000 & 0 & 1.618 & -1.618 \\ 1.000 & -1.000 & 0 & -0.6180 & 0.6180 \\ \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 zero spectrum criterion.

Data

Download links for numeric data: