The representations of a semi-direct product of a group by an abelian group , written (so is normal in ) can be described explicitly in terms of the representations of and . The purpose of this post is to explain how this is done.
Again, a good reference for all this is Serre’s well-known book, Linear representations of finite groups.
Let be a class function on $H$. Extend to trivially as follows:
for all . This is not a class function on in general. To remedy this, we “average over ” using conjugation: Define the function induced by to be
This is referred to as the Frobenius formula.
Since is normal in , acts on the vector space of formal complex linear combinations of elements of (=the characters of ),
We may restrict this action to , giving us a homomorphism , where denotes the symmetric group of all permutations of the set . This restricted action is an equivalence relation on which we refer to below as the -equivalence relation}. Let denote the set of equivalence classes of this equivalence relation. If belong to the same equivalence class then we write
(or if there is any possible ambiguity). When there is no harm, we identify each element of with a character of .
Suppose that acts on by means of the automorphism given by a homomorphism , where denotes the symmetric group of all permutations of the set . In this case, two characters are equivalent if there is an element such that, for all , we have .
For each , let
This group is called the stabilizer of in . Let
for each . There is a natural projection map
given by , i.e., by .
Extend each character from to trivially by defining
for all and . This defines a character . For each , say , let denote the representation of obtained by pulling back via the projection , i.e., define
For each and as above, let
Finally, we can completely describe all the irreducible representations of . (This is Proposition 25 in chapter 8 of Serre’s book.)
- For each and , as above, then is an irreducible representation of .
- Suppose , , . If then and .
- If is an irreducible representation of then , for some and as above.
In the next post, we will examine the special case and .