## Representation Theory of Semidirect Products: The Preliminaries (Pt. I)

**Point of Post: **In this post we begin our study of how representation theory interacts with semidirect products.

*Motivation*

As of now we have spent a considerable amount of our efforts considering how given the information (relevant to representation theory that is) about two groups and what can we say about the representation theory about certain constructions based on and . Probably the most important that we’ve so far discussed is the relationship between irreps and and the irreps of their direct product . We continue in this vein and discuss the representation theory of , the semidirect product in the particular case where is abelian.

*Semidirect Product Theory*

Let where is abelian. We recall that we can identify with and . If we write and similarly for . Consider then a representation . We next let and let, for each

Our first theorem says:

**Theorem: ***Let , and be as above. Then, is equal to .*

**Proof: **Since this quality is clearly invariant under equivalence it suffices to assume that is actually equal to

Thus, we want to prove that, letting be the representation space of the above representation,

What we claim though is that

for each in .Indeed, it’s clear that since by definition for each and each in we have that

where we’ve used the fact that any of the coordinates not corresponding to is zero. Conversely, if is such that one of its coordinates, say , is non-zero then since we may choose some such that it’s easy to see that

from where the conclusion follows. The entire theorem then follows since

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

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