Tuesday, 4 October 2011

rt.representation theory - Apocryphal Maschke theorem?

The result about bimodules is true, and standard. Here is one way to see it.



By Frobenius reciprocity, HomG(V,k[G])=Homk(V,k), since k[G] is the induction (or coinduction, depending on your terminology) of the trivial representation of the trivial subgroup of G to G.$



Since Frobenius reciprocity is functorial, one easily sees that this canonical isomorphism
is an isomorphism of right G-representations, where the source has a right G-action coming from the right G-action on k[G], and the target has a right G-action coming as the transpose of the left G-action on V.



Now if by Maschke's semisimplicity theorem, we know that
k[G]=bigoplusVVotimeskHomG(V,k[G]), where the sum is over all irreducible left G-representations. (Indeed, Mashke shows that this is true
for any left G-module in place of k[G].) Again, this is a natural isomorphism, and so respects the right G-actions on source and target.



Combined with the preceding computation, we find that
k[G]=bigoplusVVotimeskHomk(V,k)=bigoplusVEnd(V), as both left and right G-modules,
as required.



[Edit:] Leonid's remark about k being needing to be big enough in his answer below is correct. Each simple V comes equipped with an associated division algebra of G-endomorphisms
AV:=EndG(V). The representation V is absolutely irreducible (i.e stays irred. after passing to any extension field) if and only if AV=k. When we consider Homk(V,W) for another left G-module W, this is naturally an AV-module, and Maschke's theorem
will say that W=bigoplusVHomk(V,W)otimesAVV. (I have written the factors in the tensor product in this order because V is naturally a left AV-module (if we think of endomorphisms acting on the left), and then Homk(V,W) becomes a right AV-module.)



So in the case of W being the group algebra, we have
k[G]=bigoplusVHomk(V,k)otimesAVV


(an isomorphism of G-bimodules).



If all the V are absolutely irreducible, e.g. if k is algebraically closed,
then all the AV just equal k, and the preceding direct sum reduces to what I wrote above, and what was written in the question.

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