# Relation between group representations and elements of group cohomology groups

Having already seen group cohomology, I was just introduced to the formula $U \otimes Ind W = Ind(Res(U) \otimes W)$ from representation theory. This seems oddly like the formula $\mathrm{Cor}(u) \cup v = \mathrm{Cor}(u \cup \mathrm{Res}(v))$, which can be found as Proposition 1.39 in Chapter 2 of Milne's CFT Notes. Can one be proven from the other? In one case, $U, W$ are actual modules, whereas in the other case, $u,v$ are elements of modules. Maybe this means that certain $G$-modules might somehow classify representations, and the cup product would represent the tensor product of representations?

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Representations of groups correspond to $G$-equivariant vector bundles on discrete $G$-manifolds (I have written a bit on this in the comments at mathoverflow.net/questions/33828 ). Now, the $0$-th K-theory (which is a cohomology theory, after all) should classify vector bundles. If Cor and Res really correspond to Ind and Res, which I am not sure of, then we should have a relation. – darij grinberg Aug 24 '10 at 12:35