\(\text{FR}^{8,0}_{34}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{4} & \mathbf{4} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{6} & \mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{7} \ \mathbf{8})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.96432\) | \(\text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,6\right]\) |
| \(\mathbf{4}\) | \(3.82286\) | \(\text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,6\right]\) |
| \(\mathbf{5}\) | \(4.54499\) | \(\text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,6\right]\) |
| \(\mathbf{6}\) | \(5.10495\) | \(\text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,6\right]\) |
| \(\mathbf{7}\) | \(5.29385\) | \(\text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right]\) |
| \(\mathbf{8}\) | \(5.29385\) | \(\text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(128.169\) | \(\text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,6\right]^2+\text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,6\right]^2+\text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,6\right]^2+\text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,6\right]^2+2 \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right]^2+2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \hline 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,6\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,6\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,6\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,6\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,5\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,5\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,4\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,2\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,1\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,1\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,4\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,2\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,1\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,3\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,5\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,5\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,3\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,1\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,5\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,4\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,2\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,2\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,2\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,3\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,3\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,1\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,4\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,4\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,1\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,4\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,2\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,5\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,3\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,3\right] \\ 1 & -1 & 0 & 0 & 0 & 0 & 1 & -1 \\ 1 & -1 & 0 & 0 & 0 & 0 & -1 & 1 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \hline 1.000 & 1.000 & 2.964 & 3.823 & 4.545 & 5.105 & 5.294 & 5.294 \\ 1.000 & 1.000 & 2.587 & 2.104 & 0.7510 & -0.9122 & -1.555 & -1.555 \\ 1.000 & 1.000 & 1.608 & -1.023 & -2.229 & -0.3324 & 0.8475 & 0.8475 \\ 1.000 & 1.000 & 0.2300 & -2.177 & 1.446 & 1.063 & -0.6008 & -0.6008 \\ 1.000 & 1.000 & -1.068 & 0.2101 & 0.6339 & -1.521 & 0.4958 & 0.4958 \\ 1.000 & 1.000 & -1.320 & 1.063 & -1.147 & 1.597 & -0.4811 & -0.4811 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 1.000 & -1.000 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & -1.000 & 1.000 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
This fusion ring does not provide any representations of $SL_2(\mathbb{Z}).$
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: