\[((\text{SO}(8)_1) ^{\times}_{S_3})^{S_3}_e: \text{FR}^{8,0}_{29}\]
\(((\text{SO}(8)_1) ^{\times}_{S_3})^{S_3}_e: \text{FR}^{8,0}_{29}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{4}+\mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{8} & \mathbf{6}+\mathbf{7} \\ \mathbf{4} & \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{4} & \mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{6} & \mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{7} & \mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} \\ \mathbf{8} & \mathbf{8} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{4} \ \mathbf{5}), (\mathbf{6} \ \mathbf{7} \ \mathbf{8})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5}\}\) | \(\left.\text{Rep(}S_4\right):\ \text{FR}^{5,0}_{6}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.\) | \(2\) |
| \(\mathbf{4}\) | \(3.\) | \(3\) |
| \(\mathbf{5}\) | \(3.\) | \(3\) |
| \(\mathbf{6}\) | \(4.\) | \(4\) |
| \(\mathbf{7}\) | \(4.\) | \(4\) |
| \(\mathbf{8}\) | \(4.\) | \(4\) |
| \(\mathcal{D}_{FP}^2\) | \(72.\) | \(72\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 2 & 3 & 3 & 4 & 4 & 4 \\ 1 & 1 & 2 & 3 & 3 & -2 & -2 & -2 \\ 1 & 1 & 2 & -1 & -1 & 0 & 0 & 0 \\ 1 & 1 & -1 & 0 & 0 & \text{Root}\left[x^3-3 x-1,1\right] & \text{Root}\left[x^3-3 x-1,2\right] & \text{Root}\left[x^3-3 x-1,3\right] \\ 1 & 1 & -1 & 0 & 0 & \text{Root}\left[x^3-3 x-1,3\right] & \text{Root}\left[x^3-3 x-1,1\right] & \text{Root}\left[x^3-3 x-1,2\right] \\ 1 & 1 & -1 & 0 & 0 & \text{Root}\left[x^3-3 x-1,2\right] & \text{Root}\left[x^3-3 x-1,3\right] & \text{Root}\left[x^3-3 x-1,1\right] \\ 1 & -1 & 0 & 1 & -1 & 0 & 0 & 0 \\ 1 & -1 & 0 & -1 & 1 & 0 & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 2.000 & 3.000 & 3.000 & 4.000 & 4.000 & 4.000 \\ 1.000 & 1.000 & 2.000 & 3.000 & 3.000 & -2.000 & -2.000 & -2.000 \\ 1.000 & 1.000 & 2.000 & -1.000 & -1.000 & 0 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & 0 & 0 & -1.532 & -0.3473 & 1.879 \\ 1.000 & 1.000 & -1.000 & 0 & 0 & 1.879 & -1.532 & -0.3473 \\ 1.000 & 1.000 & -1.000 & 0 & 0 & -0.3473 & 1.879 & -1.532 \\ 1.000 & -1.000 & 0 & 1.000 & -1.000 & 0 & 0 & 0 \\ 1.000 & -1.000 & 0 & -1.000 & 1.000 & 0 & 0 & 0 \\ \hline \end{array}\]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
The gauging process gives a categorification of the fusion ring $\text{FR}^{8,0}{29}$ and possibly its siblings. One takes the 3-fermion theory $\text{SO}(8)_1$ and gauges a global $S_3$ symmetry, producing a rank-12, dimension-144 modular category. Taking the trivial component of its universal $\mathbf{Z}_2$ grading gives a rank 8 dimension 72 ribbon fusion category, which has fusion ring $\text{FR}^{8,0}{29}$. See refs [^1][^2] for more details.
Data
Download links for numeric data:
References
[^1] Shawn X. Cui, César Galindo, Julia Yael Plavnik & Zhenghan Wang, On Gauging Symmetry of Modular Categories. (published version and arXiv version
[^2] Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang, Symmetry Fractionalization, Defects, and Gauging of Topological Phases, (published version and arXiv version).