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\section{Induction}

\subsection{Construction}

Suppose $H$ is a subgroup of a (finite) group $G$.
Then the restriction from $G$ to $H$ gives a way
of building \glspl{rep} of $H$ from \glspl{rep} of
$G$. We want to go the other way and build
\glspl{rep} of $G$ from \glspl{rep} of $H$.

Recall that $[g]_G$ denotes the conjugacy class of
$g$ in $G$. So
\[ \indicator{[g]_G}(x) = \begin{cases}
  1 & \text{if $x$ is conjugate to $g$ in $G$} \\
  0 & \text{otherwise}
\end{cases} \]
We note that for $g \in G$,
\[ [g^{-1}]_G = [g]_G^{-1} = \{y^{-1} \st y \in
[g]_G\} \]
since $(xgx^{-1})^{-1} = xg^{-1} x^{-1}$. So
$\indicator{[g]_G}^* = \indicator{[g^{-1}]_G}$. If
$H \le G$ then $[g]_G \cap H$ is (possibly empty)
union of $H$-conjugacy classes
\[ [g]_G \cap H = \bigcup_{[h]_H \subseteq [g]_G}
[h]_H \]
\glssymboldefn{gamma_induction}{$\Gamma$}{$\Gamma$}
So $\indGamma : \mathcal{C}_G \to \mathcal{C}_H$;
$\indGamma(f) = f|_H$ is a well-defined linear map
with $\indGamma (\indicator{[g]_G}) = \sum_{[h]_H
\subseteq [g]_G} \indicator{[h]_H}$. Since for
every finite group $G$
\[ \Gip\langle f_1, f_2 \rangle_G = \frac{1}{|G|}
\sum_{g \in G} f_1^*(g) f_2(g) \]
defines a non-degenerate bilinear form on
$\mathcal{C}_G$, the map $\indGamma$ has an adjoint
$\indGamma^* : \mathcal{C}_G \to \mathcal{C}_G$ given
by
\[ \langle \indGamma(f_1), f_2 \rangle_H = \langle
f_1, \indGamma^*(f_2) \rangle_G \]
for $f_1 \in \mathcal{C}_G$, $f_2 \in
\mathcal{C}_H$. In particular for $f \in
\mathcal{C}_H$,
\[ \Gip\langle \indicator{[g^{-1}]_G}, \indGamma^*(f)
\rangle_G = \frac{1}{|G|} \sum_{x \in [g]_G}
\indGamma^*(f)(x) = \frac{1}{|C_G(g)|} \indGamma^*(f)(g) \]
On the other hand
\begin{align*}
  \Gip\langle \indicator{[g^{-1}]_G}, \indGamma^*(f)
  \rangle_G
  &= \langle \indGamma (\indicator{[g^{-1}]_G}), f
  \rangle_H \\
  &= \sum_{[h]_H \subseteq [g]_G}
  \frac{1}{|C_H(h)|} f(h)
\end{align*}
so combining these we see that
\[ \label{lec14_l60_eq} \indGamma^*(f)(g) =
\sum_{[h]_H \subseteq [g]_G}
\frac{|C_G(g)|}{|C_H(h)|} f(h) \tag{1} \]
Since $xgx^{-1} = ygy^{-1} \iff x^{-1} y \in
C_G(g)$,
\begin{align*}
  \indGamma^*(f)(g)
  &= \sum_{h \in [g] \cap H}
  \frac{|C_G(g)|}{|C_H(h)| |[h]_H|} f(h) \\
  &= \frac{1}{|H|} \sum_{x \in G} f^\circ (x^{-1}
  gx)
\end{align*}
where
\[ f^\circ(g) = \begin{cases}
  f(g) & g \in H \\
  0 & \text{otherwise}
\end{cases} \]

\textbf{Question:} Is $\indGamma^*(\charring(H))
\subseteq \charring(G)$?
Suppose $\chichar$ is a $\CC$-\gls{char_rep} of
$H$ and $\psi$ is an \gls{irred}
$\CC$-\gls{char_rep} of $G$. Then
\[ \Gip\langle \indGamma^*(\chichar), \psi \rangle_G
= \langle \chichar, \indGamma(\psi) \rangle_H \in
\NN_0 \]
by \nameref{orthog_of_chars} since $\indGamma(\psi)$
is a \gls{char_rep} of $H$.

So writing $\Irr(G) = \{\text{\gls{irred}
$\CC$-\gls{char_rep} of $G$}\}$,
\[ \label{lec14_l91_eq} \indGamma^*(\chichar) =
\sum_{\psi \in \Irr(G)} \langle \chichar, \psi|_H
\rangle_H \psi \tag{2} \]
is even a \gls{char_rep} of $G$.

\begin{example*}
  $G = S_3$ and $H = A_3 = \{e, (123), (132)\}$.
  If $f \in \mathcal{C}_H$ then by
  \eqref{lec14_l60_eq},
  \begin{align*}
    \indGamma^*(f)(e)
    &= \frac{|C_{S_3}(e)|}{|C_{A_3}(e)|} f(e)
    = \frac{6}{3} f(e) = 2f(e) \\
    \indGamma^*(f)((12))
    &= 0 \\
    \indGamma^*(f)((123))
    &= \frac{|C_{S_3}((123))|}{|C_{A_3}((123))|}
    f((123)) +
    \frac{|C_{S_3}((123))|}{|C_{A_3}((132))|}
    f((132)) \\
    &= \frac{3}{3} f((123)) + \frac{3}{3} f((132))
    = f((123)) + f((132))
  \end{align*}
  Thus
  \begin{center}
    \begin{tabular}{c|c|c|c}
      $A_3$ & $1$ & $(123)$ & $(132)$ \\
      \hline
      $\chichar_1$ & $1$ & $1$ & $1$ \\
      $\chichar_2$ & $1$ & $\omega$ & $\omega^2$ \\
      $\chichar_3$ & $1$ & $\omega^2$ & $\omega$
    \end{tabular}
    \qquad
    \begin{tabular}{c|c|c|c}
      $S_3$ & $1$ & $(12)$ & $(123)$ \\
      \hline
      $\indGamma^*(\chichar_1)$ & $2$ & $0$ & $2$ \\
      $\indGamma^*(\chichar_2)$ & $2$ & $0$ & $-1$ \\
      $\indGamma^*(\chichar_3)$ & $2$ & $0$ & $-1$
    \end{tabular}
  \end{center}
  (where $\omega = e^{2\pi i/3}$ and we use the
  fact that $\omega + \omega^{-1} = 1$).

  Thus $\indGamma^*(\chichar_1) = \indicator{} +
  \eps$ and $\indGamma^*(\chichar_2) =
  \indGamma^*(\chichar_3) = \chichar_V$ where $V$ is
  the \repdim{2} \gls{irred} \gls{rep} of $S_3$
  consisted with formula \eqref{lec14_l91_eq}
  since $\indGamma(\indicator{}) = \indGamma(\eps) =
  \chichar_1$ and $\indGamma(\chichar_V) = \chichar_2
  + \chichar_3$.
\end{example*}
\vspace{-1em}

Note that $\chichar$ is an
\gls{irred} \gls{char_rep} of $G$,
$\indGamma^*(\chichar)$ can be an \gls{irred}
\gls{char_rep} of $G$ but need not be in
general. Also note that $\indGamma^*(\chichar)(e) =
\frac{|G|}{|H|}\chichar(e)$.

We'd like to build a \gls{rep} of $G$ with
\gls{char_rep} $\indGamma^*(\chichar)$ given a
\gls{rep} $W$ of $H$ with \gls{char_rep}
$\chichar$.

\begin{flashcard}[fxw-defn]
\cloze{Suppose that $X$ is a finite set and
$W$ is a $k$-vector space we may define}
\glssymboldefn{FXW}{$F$}{$F$}
\[ \mathcal{F}(X, W) = \cloze{\{f : X \to W\}} \]
\cloze{the $k$-vector space of functions $X$ to
$W$.}
\end{flashcard}

So $\FXW(X, k) = kX$. Then $\dim
\FXW(X, W) = |X| \dim W$ since if $w_1,
\ldots, w_d$ is a basis for $W$ then
\[ (\delta_x w_i \st x \in X, 1 \le i \le d) \]
is a basis for $\FXW(X, W)$.

If $K$ is a group
and $X$ has a $K$-action and $W$ is a \gls{rep}
of $k$ then $\FXW(X, W)$ is a \gls{rep} of $K$
via $(k \cdot f)(x) = k \cdot (f(k^{-1} x))$ for
all $f \in \FXW(X, W), k \in K, x \in X$. For
example if $W = k$ is the \gls{triv_rep}, then
$\FXW(X, W) = kX$ as \glspl{rep} of $K$.

Now suppose $H \le G$ are finite groups then $G$
can be viewed as a set with $G \times H$-action
via $(g \cdot h) \cdot x = gxh^{-1}$, $\forall (g,
h) \in G \times H, x \in X$.

If $W$ is a \gls{rep} of $G$ then we can view $W$
sa a \gls{rep} of $G \times H$ via
\[ (g, h) \cdot w = h \cdot w \]
Now $\FXW(G, W)$ is a \gls{rep} of $G \times H$
via
\[ ((g, h) \cdot f)(x) = h f(g^{-1}xh) \qquad
\forall (g, h) \in G \times H, f \in \FXW(G, W), x
\in G \]
So it can be viewed as a \gls{rep} of $G$ and as a
\gls{rep} of $H$ via $g \mapsto (g, e_H)$ and $h
\mapsto (e_G, h)$ respectively and the actions
commute.

Now
\begin{align*}
  \FXW(G, W)^H
  &= \{f \in \FXW(G, W) \st (e, h) f = f ~\forall
  h \in H\} \\
  &= \{f \in \FXW(G, W) \st f(xh) = h^{-1} f(x)
  ~\forall h \in H, x \in G\}
\end{align*}
is a \Ginv{} subspace of $\FXW(G, W)$ since if
$(e, h) \cdot f = f$ then for $g \in G$,
\[ (e, h)(g, e) f = (g, e)(e, h) f = (g, e) f \]

\begin{example*}
  $\FXW(G, k)^H \simeq k G / H$ if $k$ is the
  \gls{triv_rep} of $G$.
\end{example*}

\begin{flashcard}[induced-rep-defn]
\begin{definition*}[Induced representation]
  \glsverbdefn{induction}{induction}{N/A}
  \glssymboldefn{Ind}{Ind${}_H^G$}{Ind${}_H^G$}
  \cloze{
  Suppose $H$ is a subgroup of a finite group $G$
  and $W$ is a \gls{rep} of $H$. The \emph{induced
  representation}
  \[ \Ind^G_H W = \FXW(G, W)^H \]
  is a \gls{rep} of $G$.
  }
\end{definition*}
\end{flashcard}

\begin{flashcard}[dim-induced-rep-lemma]
\begin{lemma*}
  $\dim \Ind_H^G W = \cloze{\frac{|G|}{|H|} \dim W}$.
\end{lemma*}

\begin{proof}
  \cloze{
  Let $X = G / H$ be the left cosets of $H$ in $G$
  and let $x_1, \ldots, x_{|G / H|}$ be coset
  representatives. Then
  \begin{align*}
    \theta : \FXW(G, W)^H
    &\to \FXW(X, W) \\
    \theta(f) (x_i H)
    &= f(x_i)
  \end{align*}
  is a $k$-linear map with inverse
  $\varphi(l)(x_i h) = h^{-1} l(x_i)$ for all $l
  \in \FXW(X, W), h \in H$, $i = 1, \ldots,
  |G/H|$.
  }
\end{proof}
\end{flashcard}

\begin{theorem*}[Frobenius reciprocity]
  \label{frob_recip}
  If $V$ is a \gls{rep} of $G$ and $W$ is
  \gls{rep} of $G$ then
  \[ \GHom_G(V, \Ind_H^G W) \simeq
  \GHom_H(\Res_H^G V, W) \]
  \glssymboldefn{Res}{Res}{Res}
  where $\Res_H^G V$ is the restriction of $V$ to
  $H$.
\end{theorem*}