Why are there so few quaternionic representations of simple groups? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:50:06Z http://mathoverflow.net/feeds/question/54800 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54800/why-are-there-so-few-quaternionic-representations-of-simple-groups Why are there so few quaternionic representations of simple groups? ARupinski 2011-02-08T18:25:59Z 2011-02-08T18:25:59Z <p>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.</p> <p><b>Question:</b> 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)? </p>