# Abstract Nonsense

## 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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Second Formula

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Let $G$ be a finite group, $H\leqslant G$, and $\rho:H\to\mathcal{U}\left(\mathscr{V}\right)$ a representation. Consider then the first form of the induced representation: $\text{Ind}^G_H(\rho):G\to\mathcal{U}\left(\mathscr{X}\right)$. If $\chi$ is the character for $\rho$ we define, analogously to the previous post, the induced character $\text{Ind}^G_H(\chi)$ to be the character of $\text{Ind}^G_H(\rho)$. We now compute $\text{Ind}^G_H(\chi)$ in terms of $\chi$. But first we make an observation:

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Lemma: Let $G$ be a group and $C$ a conjugacy class in $G$. Then, for any $H\leqslant G$ there exists distinct conjugacy classes $C_1,\cdots,C_n$ in $H$ such that $C\cap H=C_1\cup\cdots\cup C_n$.

Proof: Indeed we let $C_1,\cdots,C_n$ be the elements of $\left\{C_h:h\in C\cap H\right\}$ where $C_h$ is the conjugacy class in $H$ of $h$. Clearly then for each $h\in C\cap H$ we have that $C_h\subseteq C\cap H$ (since $C\cap H=\left\{ghg^{-1}:g\in G\right\}\cap H\supseteq\left\{khk^{-1}:k\in H\right\}=C_h$) and so $C_1\cup\cdots\cup C_n\subseteq C\cap H$. Conversely, if $h\in C\cap H$ then $h\in C_h$ and so $\displaystyle C\subseteq \bigcup_{h\in H}C_h=C_1\cup\cdots\cup C_n$. $\blacksquare$

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Moreover, it’s clear that this decomposition is unique. Thus, we may state unambiguously:

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Theorem: Let $G$ be a finite group, $H\leqslant G$, and $\rho:H\to\mathcal{U}\left(\mathscr{V}\right)$ a representation of $H$ with character $\chi$. Consider then the first construction of the induced representation $\text{Ind}^G_H(\rho):G\to\mathcal{U}\left(\mathscr{X}\right)$. Then, if $C$ is any conjugacy class and $C\cap H$ has the unique decomposition (as described in the lemma) as $C\cap H=C_1\cup\cdots\cup C_n$ where $C_j$ is a conjugacy class in$H$ then

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$\displaystyle \text{Ind}^G_H(\chi)(C)=\frac{(G:H)}{\#(C)}\sum_{j=1}^{n}\#(C_j)\chi(C_j)$

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(where we recall that since a character is a class function by $\text{Ind}^G_H(\chi)(C)$ and $\chi(C_j)$ we mean the unique value each of those characters takes on the conjugacy class).

Proof: We begin by considering the usual extension of $\text{Ind}^G_H(\rho)$ to $\mathcal{A}(G)$ given by

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$\displaystyle \varrho:\mathcal{A}(G)\to\text{End}\left(\mathscr{X}\right):a\mapsto \sum_{g\in G}a(g)\text{Ind}^G_H(\rho)_g$

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and recall that $\varrho$ is in fact a homomorphism of $\ast$-algebras. Then we see that

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\displaystyle \begin{aligned}\text{tr}\left(\varrho\left(\sum_{x\in C}\delta_ x\right)\right) &=\sum_{x\in C}\sum_{g\in G}\text{tr}\left(\delta_x(g)\text{Ind}^G_H(\rho)_x\right)\\ &=\sum_{x\in C}\text{tr}\left(\text{Ind}^G_H(\rho)_x\right)\\ &=\sum_{x\in C}\text{Ind}^G_H(\chi)(C)\\ &=\#(C)\text{Ind}^G_H(\chi)(C)\end{aligned}

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Let then $v_1,\cdots,v_m$ be a basis for $\mathscr{V}$ and $\{t_1,\cdots,t_r\}$ a transversal for $G/H$. Define then $f_j$ for $j=1,\cdots,m$ by

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$\displaystyle f_j(g)=\begin{cases}\sqrt{(G:H)}\rho_g^{-1}(v_j) & \mbox{if}\quad g\in H\\ \bold{0} & \mbox{if}\quad g\notin H\end{cases}$

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what we claim then is that if $f_{i,j}=\text{Ind}^G_H(\rho)_{t_i}\left(f_j\right)$ for $i\in[r]$ and $j\in[m]$ then $\left\{f_{i,j}\right\}_{i\in[r],j\in[m]}=\mathcal{B}$ is an orthonormal basis for $\mathscr{X}$. Since $(G:H)\text{dim}(\mathscr{V})=\#\left(\mathcal{B}\right)$ which we know is equal to $\dim\left(\mathscr{X}\right)$ it clearly suffices to prove orthonormality. To do this we note that

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\displaystyle \begin{aligned}\left\langle f_{i,j},f_{p,q}\right\rangle &= \frac{1}{|G|}\sum_{g\in G}\left\langle f_{i,j}(g),f_{p,q}(g)\right\rangle_\mathscr{V}\\ &= \frac{1}{|G|}\sum_{g\in G}\left\langle \text{Ind}^G_H(\rho)_{t_i}(f_j)(g),\text{Ind}^G_H(\rho)_{t_p}(f_q)(g)\right\rangle_{\mathscr{V}}\\ &= \frac{1}{|G|}\sum_{g\in G}\left\langle f_j\left(t_i^{-1}g\right),f_q\left(t_p^{-1}g\right)\right\rangle_{\mathscr{V}}\end{aligned}\quad\quad\mathbf{(1)}

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References:

1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.