# Abstract Nonsense

## 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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Frobenius Reciprocity Theorem

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We waste no time and proceed with the theorem. The most beautiful thing about this theorem though is how simple the proof is–it’s literally just a simple computation, but it is very, very powerful.

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Theorem (Frobenius Reciprocity): Let $G$ be a finite group and $H\leqslant G$. Then, the two maps $\text{Ind}^G_H:\text{Cl}(H)\to\text{Cl}(G)$ and $\text{Res}^H_G:\text{Cl}(G)\to\text{Cl}(H)$ are adjoint. In other words, for any $f_1\in\text{Cl}(H)$ and $f_2\in\text{Cl}(G)$

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$\displaystyle \left\langle f_1,\text{Res}^H_G(f_2)\right\rangle_{\text{Cl}(H)}=\left\langle \text{Ind}^G_H(f_1),f_2\right\rangle_{\text{Cl}(G)}$

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Proof: Let $f_1\in\text{Cl}(H)$ and $f_2\in\text{Cl}(G)$ be arbitrary and we recall that if we fix some transveral $\mathscr{T}$ for $G/H$ then

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$\displaystyle \text{Ind}(f)(g)=\sum_{g\in G}f^\circ\left(t^{-1}gt\right)=\frac{1}{|H|}\sum_{k\in G}f^\circ\left(k^{-1}gk\right)$

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where $f^\circ$ is zero on $G-H$ and $f$ on $H$. Then, by definition one has

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\displaystyle \begin{aligned}\left\langle \text{Ind}^G_H(f_1),f_2\right\rangle_{\text{Cl}(G)} &= \frac{1}{|G|}\sum_{g\in G}\text{Ind}^G_H(f_1)(g)\overline{f_2(g)}\\ &= \frac{1}{|G||H|}\sum_{g\in G}\left(\sum_{k\in G}f_1^\circ\left(k^{-1}gk\right)\right)\overline{f_2(g)}\\ &= \frac{1}{|G||H|}\sum_{k\in G}\sum_{g\in G}f_1^\circ\left(k^{-1}gk\right)\overline{f_2(g)}\\ &= \frac{1}{|G||H|}\sum_{k\in G}\sum_{h\in H}f_1^\circ(h)\overline{f_2\left(khk^{-1}\right)}\\&= \frac{1}{|G||H|}\sum_{k\in G}\sum_{h\in H}f_1(h)\overline{f_2(h)}\\ &= \frac{1}{|H|}\sum_{h\in H}f_1(h)\overline{f_2(h)}\\ &= \left\langle f_1,\text{Res}^H_G\left(f_2\right)\right\rangle_{\text{Cl}(H)}\end{aligned}

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where we used the obvious fact that the only places where $k^{-1}gk\in H$ is precisely $kHk^{-1}$ and summing over this is precisely summing over $H$ where we replace $g$ by $khk^{-1}$–and the fact that $f_2$ is a class function on $G$ so that $f_2\left(khk^{-1}\right)=f_2(h)$ for every $h\in H$. $\blacksquare$

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Note that if we were doing character theory and solely interested in the characters of $G$ and not the representations then we could have defined the induced map $\text{Ind}^G_H$ to be equal to the unique adjoint of $\text{Res}^H_G$ and then defined the induced character of a character $\chi$ in $H$ to be $\text{Ind}^G_H(\chi)=\bigtriangleup(\chi)$ the fact that it would be character follows from the following:

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Multiplicities of Irreps

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We now discuss a fantastic corollary of the Frobenius Reciprocity theorem. The beginnings of which come from the offhand remark at the end of the above section. Namely that it somehow follows from the Frobenius Reciprocity theorem that $\bigtriangleup(\chi)$ is a character of $G$ whenever $\chi$ is a character of $H$. We’ll recall that this reduces to showing that $\bigtriangleup(\chi)\in\text{span}_{\mathbb{N}}(\text{irr}(G))$ or that $\left\langle \bigtriangleup(\chi),\mu\right\rangle_{\text{Cl}(G)}\in\mathbb{N}$ for every $\mu\in\text{irr}(G)$. But, it’s trivial that $\text{Res}^H_G(\mu)$ is a character of $H$ (it’s the character of the restriction of $\rho$ to $\rho_{\mid H}$ if $\rho$ is the character for $\mu$) and so $\left\langle \chi,\text{Res}^H_G(\mu)\right\rangle_{\text{Cl}(H)}\in\mathbb{N}$ from where the rest follows from Frobenius Reciprocity theorem. We’d now like to extend this observation to show a brilliant fact about the multiplicities of an induced representation. Namely, for a group $K$ and a representation $\rho$ on $K$ we define, for each $\alpha\in\widehat{G}$, the multiplicity of $\alpha$ in $K$ denoted $\text{mult}\left(\rho,\alpha\right)$ to be the number of times an element of $\alpha$ occurs in the decomposition of $\rho$ into irreps. But, it’s trivial using the orthogonality of irreducible characters to note that $\displaystyle \text{mult}\left(\rho,\alpha\right)=\left\langle\rho,\chi^{(\alpha)}\right\rangle$. The brilliant thing though is that given two irreps$\rho^{(\alpha)}$ of $G$ and $\psi^{(\beta)}$ of $H$ we see that

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$\text{mult}\left(\text{Ind}^G_H(\psi^{(\beta)}),\alpha\right)=\left\langle \bigtriangleup\left(\chi^{(\beta)}\right),\chi^{(\alpha)}\right\rangle=\left\langle \chi^{(\beta)},\bigtriangledown\left(\chi^{(\alpha)}\right)\right\rangle=\text{mult}\left(\text{Res}^H_G\left(\rho^{(\alpha)}\right),\beta\right)$

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Which, when you think about it, is a crazy thing. The number of times something in $\alpha$ occurs in the decomposition of an induced representation of something in $\beta$ is equal to the number of times something in $\beta$ appears in the decomposition of the restriction. Just amazing.

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References:

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