Timeline for Irreducible representations of a product of two groups

3 events

when toggle format	what		by	license	comment
Dec 11, 2022 at 15:41	comment	added	Benjamin Steinberg		Actually I think your proof is fine for infinite groups with a tiny modification. You can argue that the restriction of an irreducible to a direct factor (or any normal subgroup) is completely reducible by noting the sum of all the translates of an irreducible subrep of the normal subgroup is the whole representation by irreduciblity and then proceed as in your proof. I'll keep mine since it works for monoids or for any algebras
Dec 11, 2022 at 14:58	comment	added	Benjamin Steinberg		The OP doesn't say that the group here is finite. In fact he may be looking at topological irreducibility which for compact groups is the same and I guess this is true in finite dimensions for all topological groups since all subspaces are closed but maybe one has to check continuity. Anyway for reps of discrete infinite groups you need to argue differently because you don't get for free complete reducibility but it isn’t bad. Nonetheless is true as I argue in the comments. The argument for finite dimensional algebras is basically the same as your note after you factor out the radicals.
Dec 11, 2022 at 14:24	history	answered	Andy Putman	CC BY-SA 4.0