Say I have an unknown group $G$ with a simple, real, faithful representation $\mathbf n$ such that $$ {\mathbf n}\otimes{\mathbf n} \approx {\mathbf 1}_s\oplus{\mathbf a}_s\oplus{\mathbf b}_s\oplus{\mathbf c}_a$$ for simple representations $\bf a$, $\bf b$, and $\bf c$ of known dimensions. How can I find groups $G$ with this property? Under what conditions can such a group exist?
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1$\begingroup$ 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. $\endgroup$– Noah SnyderCommented Aug 8, 2019 at 19:36
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$\begingroup$ 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? $\endgroup$– djbinderCommented Aug 8, 2019 at 20:08
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$\begingroup$ 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). $\endgroup$– Noah SnyderCommented Aug 8, 2019 at 20:21
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$\begingroup$ 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. $\endgroup$– Noah SnyderCommented Aug 8, 2019 at 20:36
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$\begingroup$ My techniques also work for braided tensor categories, since groups are such a special case there may be group-specific techniques that work better. $\endgroup$– Noah SnyderCommented Aug 8, 2019 at 20:38
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