Timeline for Determining the groups compatible with given fusion rules
Current License: CC BY-SA 4.0
10 events
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| Aug 8, 2019 at 23:17 | comment | added | Noah Snyder | Ah, sorry that's my mistake, I hadn't understood your subscripts. Yes, for $G_2$ the copy of the $7$-dimensional is anti-symmetric, there's no clever way to change that. So depending on which factor is $n$ you're either in the symmetric group (and friends) case or the $G_2$ (and friends) case. | |
| Aug 8, 2019 at 22:42 | comment | added | djbinder | One final question, $G_2$ has the fusion rule ${\bf 7}\otimes{\bf 7}\rightarrow {\bf 1}_s + {\bf 28}_s+{\bf 7}_a+{\bf 14}_a$ so the trivalent vertex is antisymmetric not symmetric. Is this what you mean, or is there some clever way to fix this so that the vertex is symmetric? | |
| Aug 8, 2019 at 22:01 | comment | added | Noah Snyder | $S_n$ isn't "trivalent" because it is not generated by the trivalent vertex. Instead it has a trivalent subcategory, which must have smaller 4-box space, and hence must be quantum SO_3. You can then use an additional argument (not yet in print, though half of it appears in these notes (pages.iu.edu/~nsnyder1/macalaster.pdf) to show that the category generated by the trivalent vertex and the crossing has to be Deligne's $S_t$. | |
| Aug 8, 2019 at 21:33 | comment | added | djbinder | Thanks for the reference! I'm confused though about $\mathsf{Rep}(S^n)$, as this seems to contradict Corollary 8.9 unless I've misunderstanding something. | |
| Aug 8, 2019 at 20:38 | comment | added | Noah Snyder | My techniques also work for braided tensor categories, since groups are such a special case there may be group-specific techniques that work better. | |
| Aug 8, 2019 at 20:36 | comment | added | Noah Snyder | The main techniques for that case with the argument I have in mind are in Morrison-Penneys-S. arxiv.org/abs/1501.06869, but you need to do some nontrivial additional work beyond what's there. This is more a "I'm confident I could advise a grad student through this question" situation than a "I could explain to you exactly what the answer is and how to do it" situation. | |
| Aug 8, 2019 at 20:21 | comment | added | Noah Snyder | Also the highly transitive subgroups of symmetric groups (alternating and Mathieau) plus G2 (and possibly highly transitive subgroups of G2, but I don't know if someone's worked out that classification for G2). | |
| Aug 8, 2019 at 20:08 | comment | added | djbinder | For the special case where ${\bf a}\approx {\bf n}$ what are you able to say? I know the fundamental of $S^{n+1}$ works (and obviously so does $\mathbb Z_2 \times S^{n+1}$), are there other examples? | |
| Aug 8, 2019 at 19:36 | comment | added | Noah Snyder | If you only had two nontrivial summands, or even if you knew that one of the trivial summands was again n, then I could answer your question. But this question is just beyond the techniques I know. | |
| Aug 8, 2019 at 18:00 | history | asked | djbinder | CC BY-SA 4.0 |