## Mackey Irreducibility Criterion

**Point of Post: **In this post we prove the Mackey Irreducibility Criterion.

*Motivation*

In this post we put to rest probably one of the most niggling questions about induced representations. Namely, we have so far figured out how to take representations on and create a representation on but have yet to give a condition for which the a property we are so fond of us is true. To be direct, we have yet to decide when is actually an irrep of . The idea is simple, namely we know that being an irrep is equivalent to having but from Frobenius Reciprocity theorem we know that this is equivalent to showing the slightly more scary looking . But, thanks to Mackey we have information on this right hand term.

## Induced Representation is ‘Inductive’

**Point of Post: **In this post we use the Frobenius Reciprocity theorem to show that the construction of Induced representation is inductive, in the sense that for any .

*Motivation*

An obvious question is the following: given groups there are two ways we can create a representation on from a representation on . Namely, we take a representation of of we can then obviously consider the induced representation , but we could also consider creating a -representation by considering and then successively considering the -representation . The obvious question though is that if we do this, do we get two different representation. The answer turns out to be no, in other words it’s true that given any -representation the representations and are the same representation. In general this may be a very messy task, but thanks to the Frobenius Reciprocity theorem this becomes almost obvious in the sense that since is the adjoint of it suffices to show that the map is the adjoint of and we will be done by the uniqueness of the adjoint. The fact that this is true is often stated that inducing representations is *inductive.*

## Frobenius Reciprocity

**Point of Post: **In this post we discuss the Frobenius Reciprocity Theorem and discuss some of its consequences including its obvious relation to multiplicities.

*Motivation*

In our last post we saw that given a finite group and some there is a natural map by extending linearly the map for every where is the induced character. We saw dually there was a map which just took a class function on and restricted it to . In this post we prove the amazing fact that the maps and are adjoint, in the sense that . What of course then shall be true is that the maps and are adjoint. From this we shall derive a fascinating result about the relation between how often an irrep occurs in the decomposition of and how often it occurs in .

## Induced Class Functions and the Space of Integral Class Functions (Pt. II)

**Point of Post: **This is a continuation of this post.

## Induced Class Functions and the Space of Integral Class Functions (Pt. I)

**Point of Post: **In this post we set up the frame work to say the Frobenius reciprocity theorem intelligently. Namely, we discuss the space of all linear characters of a group and describe the natural maps and .

*Motivation*

The beginnings of the theory in this post can be traced back to our first derivation for the induced character of an induced representation when one noticed that by extending every (irreducible) character on to a character (thus a class function) we’ve automatically defined a map by extending the map by linearity (recalling that the irreducible characters form a basis for ). But, what we’ll note that is if we restrict to the space we get that . Thus when we restrict to we get a map . The subject of this post will be to explore theses topics and their dual concepts–namely the obviously defined map and .

## Different Formula For the Character of an Induced Representation (Pt. III)

**Point of Post: **This post is a continuation of this one.

## Different Formula For the Character of an Induced Representation (Pt. II)

**Point of Post: **This post is a continuation of this one.

## Different Formula For the Character of an Induced Representation (Pt. I)

**Point of post: **In this post we derive a formula for the induced character of a representation different than the one derived in the previous post.

** Motivation** In our last post we derived a formula for the induced character of a representation. The interesting thing is that we derived this formula using the ‘type’ of induced representation we derived second. A natural question is that if we can use our first equivalent form of induced representations to derive another formula for induced characters. In this post we do precisely that.

## The Character of an Induced Representation

**Point of Post:** In this post we discuss the notion of the induced character of an induced representation and provide a formulaic relationship between the induced character and the original character.

*Motivation*

As we have already seen given a finite group a subgroup and a representation we can create a representation . Of course though, representations aren’t just the main focus in representation theory, we are also heavily interested in the representations characters since they holds so much information about the representations they come from as well as the underlying group as well. Thus, in this post we derive a relationship between the character of a representation of and the character of the associated induced representation for .

## Another Way of Looking at Induced Representations (Pt. II)

**Point of Post:** This post is a continuation of this one.