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Divisibility theorems for group representations II October 14, 2009

Posted by Akhil Mathew in algebra, representation theory.
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2 comments

So last time we proved that the dimensions of an irreducible representation divide the index of the center. Now to generalize this to an arbitrary abelian normal subgroup.

There are first a few basic background results that I need to talk about. 

Induction  

Given a group {G} and a subgroup {H} (in fact, this can be generalized to a non-monomorphic map {H \rightarrow G}), a representation of {G} yields by restriction a representation of {H}. One obtains a functor {\mathrm{Res}^G_H: Rep(G) \rightarrow Rep(H)}. This functor has an adjoint, denoted by {\mathrm{Ind}_H^G: Rep(H) \rightarrow Rep(G)}. (more…)