# Can the defining rep of $E_7$ split over a finite subgroup while the adjoint remains simple?

Does the (simply connected compact) Lie group $$E_7$$ contain a finite subgroup $$G \subset E_7$$ such that the $$56$$-dimensional irrep of $$E_7$$ splits over $$G$$ as $$28 \oplus \overline{28}$$, but the $$133$$-dimensional adjoint representation remains simple when restricted to $$G$$? By "$$28 \oplus \overline{28}$$" I mean of course a $$28$$-dimensional complex irrep plus its dual.

Such a subgroup is definitely Lie primitive (meaning it doesn't fit inside a proper Lie subgroup $$L \subset E_7$$), since an imprimitive subgroup would split $$133 = \mathfrak{l} \oplus (\dots)$$. According to Griess and Ryba - Finite simple groups which projectively embed in an exceptional Lie group are classified!, of the (quasi)simple subgroups of $$E_7$$, only three are primitive, and none of them work. ($$133$$ splits over $$SL_2(29)$$ and $$SL_2(37)$$, and $$56$$ remains simple over $$PSU_3(8)$$.) But I don't know about nonsimple subgroups.

This feels like a good homework problem, but in fact came up in my research: such a subgroup would allow me to build a superconformal field theory with some nice properties.

• I love the title of that paper. I know Bob as a very calm person, so can only imagine how much the result must have excited him for him to put that exclamation mark. – LSpice Jun 22 '19 at 2:01
• I don't think this is homework problem. I haven't worked out all details, and I will probably write up some sort of answer. I think the difficult case might be if the $28$-dimensional irreducible representation is induced from $1$-dimensional representation of a subgroup of index $28$. – Geoff Robinson Jun 22 '19 at 17:43
• @GeoffRobinson I take it that you think that there is no such $G$ (which also seems most likely to me). I look forward to your write up. – Theo Johnson-Freyd Jun 23 '19 at 16:17
• I am not certain at present that there is no such $G$, though it would be my inclination to believe that. I am currently trying to prove that such a $G$ would have a quasisimple subnormal subgroup acting irreducibly on the $28$-dimensional module and on the $133$-dimensional module, which you have essentially already excluded. If the $28$-dimensional rep of $G$ is not monomial, I think I can do it. If it is monomial, there seems to be more to be done. – Geoff Robinson Jun 23 '19 at 16:46

I know this is an old question, but I can answer it in the negative. First, I have mostly completed a list of the Lie primitive subgroups of $$E_7(k)$$ for all $$k$$, including $$k=\mathbb C$$, and there is no such example.
In fact, for $$p=5$$ there are examples, namely the sporadic groups $$M_{22}$$, $$HS$$ and $$Ru$$. But there is none over $$\mathbb{C}$$.