Abstract Nonsense

Mackey Irreducibility Criterion

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

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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 $H\leqslant G$ and create a representation on $G$ 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 $\text{ind}^G_H(\psi)$ is actually an irrep of $G$. The idea is simple, namely we know that being an irrep is equivalent to having $\left\langle \text{ind}^G_H(\chi_\psi),\text{ind}^G_H(\chi_\psi)\right\rangle=1$ but from Frobenius Reciprocity theorem we know that this is equivalent to showing the slightly more scary looking   $\left\langle \chi_\psi,\text{Res}^H_G\left(\text{ind}^G_H(\chi_\psi)\right)\right\rangle=1$. But, thanks to Mackey we have information on this right hand term.

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May 6, 2011

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 $\text{Ind}^H_K\circ\text{Ind}^G_H=\text{Ind}^G_K$ for any $K\leqslant H\leqslant G$.

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Motivation

An obvious question is the following: given groups $K\leqslant H\leqslant G$ there are two ways we can create a representation on $G$ from a representation on $K$. Namely, we take a representation of $\rho$ of $K$ we can then obviously consider the induced representation $\text{Ind}^G_K(\rho)$, but we could also consider creating a $H$-representation by considering $\text{Ind}^H_K(\rho)$ and then successively considering the $G$-representation $\text{Ind}^G_H\left(\text{Ind}^H_K(\rho)\right)$. 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 $K$-representation $\rho$ the representations $\text{Ind}^G_K(\rho)$ and $\text{Ind}^G_H\left(\text{Ind}^H_K(\rho)\right)$ 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 $\text{Ind}^G_K:\text{Cl}(K)\to\text{Cl}(G)$ is the adjoint of $\text{Res}^K_G:\text{Cl}(G)\to\text{Cl}(K)$ it suffices to show that the map $\text{Ind}^G_H\circ\text{Ind}^H_K:\text{Cl}(K)\to\text{Cl}(G)$ is the adjoint of $\text{Res}^K_G$ 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.

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May 1, 2011

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.

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Motivation

In our last post we saw that given a finite group $G$ and some $H\leqslant G$ there is a natural map $\text{Cl}(H)\to\text{Cl}(G)$ by extending linearly the map $\displaystyle \chi\mapsto \text{Ind}^G_H(\chi)$ for every $\chi\in\text{irr}(H)$ where $\text{Ind}^G_H(\chi)$ is the induced character. We saw dually there was a map $\text{Res}^H_G:\text{Cl}(G)\to\text{Cl}(H)$ which just took a class function on $G$ and restricted it to $H$. In this post we prove the amazing fact that the maps $\text{Ind}^G_H$ and $\text{Res}^H_G$ are adjoint, in the sense that $\left\langle f_1,\text{Res}^H_G(f_2)\right\rangle_{\text{Cl}(H)}=\left\langle\text{Ind}^G_H(f_1),f_2\right\rangle_{\text{Cl}(G)}$. What of course then shall be true is that the maps $\bigtriangleup$ and $\bigtriangledown$ are adjoint. From this we shall derive a fascinating result about the relation between how often an irrep occurs in the decomposition of $\rho$ and how often it occurs in $\text{Ind}^G_H(\rho)$.

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May 1, 2011

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

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

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April 27, 2011 Posted by | Algebra, Representation Theory | , , , , , | Comments Off on Induced Class Functions and the Space of Integral Class Functions (Pt. II)

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 $\mathbb{Z}\left(\text{irr}(G)\right)$ of a group and describe the natural maps $\bigtriangleup:\mathbb{Z}\left(\text{irr}(H)\right)\to\mathbb{Z}\left(\text{irr}(G)\right)$ and $\bigtriangledown:\mathbb{Z}\left(\text{irr}(G)\right)\to\mathbb{Z}\left(\text{irr}(H)\right)$.

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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 $H$ to a character (thus a class function) we’ve automatically defined a map $\text{Ind}^G_H:\text{Cl}\left(H\right)\to\text{Cl}\left(G\right)$ by extending the map $\chi\mapsto \text{Ind}^G_H(\chi)$ by linearity (recalling that the irreducible characters form a basis for $\text{Cl}(H)$). But, what we’ll note that is if we restrict $\text{Cl}(H)$ to the space $\text{span}_{\mathbb{Z}}(\text{irr}(H))\overset{\text{def.}}{=}\mathbb{Z}\left(\text{irr}(H)\right)$ we get that $\text{Ind}^G_H\left(\mathbb{Z}\left(\text{irr}(H)\right)\right)\subseteq\mathbb{Z}\left(\text{irr}(G)\right)$. Thus when we restrict $\text{Ind}^G_H$ to $\mathbb{Z}\left(\text{irr}(H)\right)$ we get a map $\bigtriangleup:\mathbb{Z}\left(\text{irr}(H)\right)\to\mathbb{Z}\left(\text{irr}(G)\right)$. The subject of this post will be to explore theses topics and their dual concepts–namely the obviously defined map $\text{Res}^H_G:\text{Cl}(G)\to\text{Cl}(H)$ and $\bigtriangledown:\mathbb{Z}\left(\text{irr}(G)\right)\to\mathbb{Z}\left(\text{irr}(H)\right)$.

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April 27, 2011

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

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

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April 26, 2011

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

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

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April 26, 2011

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.

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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.

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April 26, 2011

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.

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Motivation

As we have already seen given a finite group $G$ a subgroup $H\leqslant G$ and a representation $\rho:H\to\mathcal{U}\left(\mathscr{V}\right)$ we can create a representation $\text{Ind}^G_H(\rho):G\to\mathcal{U}\left(\mathscr{X}\right)$. 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 $H$ and the character of the associated induced representation for $G$.

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April 25, 2011

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

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

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April 24, 2011