The suspension spectrum of a group can be thought of as a kind of group ring S[G]. It would be interesting to try to see how many of the classical results about Hopf-algebras (or just group rings) one could obtain for such objects.
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In other words, study "representation theory" for groups over the sphere spectrum S instead of any field.
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But that brings to mind a somewhat simpler problem I suppose: do people do representation theory over the integers? Is it too hard?
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Replying to @koszuldude
(I realize that's not entirely in the sense you meant, but I wanted to point out that we get to modular forms here as well)
10:09 PM - 22 Feb 2019
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