So I’ve thrown out lots of terms and theorems relating to representation theory, but haven’t really said how any of it relates, so lets try to start putting a few things together before moving on. I threw out the terms reducible and irreducible, but forgot to mention completely reducible. A representation is completely reducible if it can be decomposed as a direct sum of irreducible subrepresentations. Note that this means an irreducible representation is completely reducible.
Let’s start with finite abelian groups where everything is easier. Then we’ll eventually get to the full theory of general finite groups, where essentially all the same statements will hold, but I’ll need bigger theorems and techniques.
I claim that every complex representation of a finite abelian group is completely reducible, and in a very nice way. So by Schur’s Lemma, we get that any irreducible subrepresentation is going to be one-dimensional, since group elements will be acting by scalars. So it will leave lines invariant, and hence I will be looking to decompose into lines somehow.
This becomes fairly clear on how to actually decompose when you think about the fact that matrices that are diagonalizable and commute are actually simultaneously diagonalizable. So let G be a finite abelian group. And diagonalizing is the same thing as the intertwiners we saw before, so any representation of G is isomorphic to one where every group element acts diagonally. So our representation where , and the are lines. Each line is preserved by the action of the group.
Now just use the structure theorem for abelian groups, i.e. that G is isomorphic to a direct sum of cyclic subgroups. Thus given , the actions on a line can be thought of as the n roots of unity as our choices for the element of .
Thus every complex representation of a finite abelian group is completely reducible into 1-dimensional subrepresentations (this will be a special case of something we do later).