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Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I recall, and only 2 out of its 24 irreps are quaternionic) has any irreps of quaternionic type. On the other hand, to my recollection several members of infinite families (such as those arising from the symplectic groups) as well as certain covers of the sporadics have quaternionic irreps. As I do not currently have access to the Atlas, I can't really list a bunch of examples, but if you have access to a copy you can go look them up.

Question: Is there a 'natural' reason that quaternionic representations and simple groups (in particular sporadics) like to avoid one another? Specifically, is there something intrinsic about preserving a symplectic form which implies that the corresponding automorphism group "should" have a normal subgroup (because of something trivial like symmetry considerations)?

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    $\begingroup$ My guess is something like "it's hard for large 2-groups to have symplectic representations, so any simple group with one must have low 2-rank." $\endgroup$
    – Ben Webster ♦
    Commented Feb 8, 2011 at 19:15
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    $\begingroup$ It looks like a lot of unitary groups and a lot of twisted groups in general have quaternionic irreducible ordinary representations. It might not be rare, and doesn't seem like it requires low 2-rank. $\endgroup$
    – Jack Schmidt
    Commented Feb 9, 2011 at 0:05
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    $\begingroup$ Srinivasan–Vinroot have a preprint showing unitary groups of even degree have a ton of quaternionic irreps, and gives a reason why. math.wm.edu/~vinroot/SemiSympChars-REV.pdf $\endgroup$
    – Jack Schmidt
    Commented Feb 9, 2011 at 0:09
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    $\begingroup$ @Jack: From your comments and the sharp counts of quaternionic representations in the S-V paper, it seems plausible that if one starts with symplectic forms and tries to determine which finite simple groups can leave such forms invariant, one would eventually determine some families of representations of classical groups and two sporadic representations (of the McLaughlin Group). Thus it looks like my question would be equivalent to a classification problem and hence likely difficult. $\endgroup$
    – ARupinski
    Commented Feb 9, 2011 at 7:42
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    $\begingroup$ That is an interesting question, which I've thought about in the context of Lie groups. If G is a compact connected Lie group then the real/quaternionic indicator (equal to the Frobenius-Schur indicator) of a representation is its central character evaluated at a certain element. So (for example) if G is adjoint then every irreducible self-dual representation is real, and in general "at most half" of representations are quaternionic. This is more of an observation than an explanation. See arxiv.org/abs/1203.1901 $\endgroup$
    – Jeffrey Adams
    Commented Jun 6, 2016 at 20:41

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