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\subsection{Orthogonality of characters}

Assume $G$ is always a finite group and $k = \CC$.

Recall
\[ \mathcal{C}_G \defeq \{f : G \to \CC :
f(hgh^{-1}) = f(g) ~\forall g, h \in G\} \le \CC G \]
and if $O_1, \ldots, O_r$ are the conjugacy
classes then the indicator functions
$\indicator{O_1}, \ldots, \indicator{O_r}$ are a
basis.

\glssymboldefn{Gip_class_funcs}{G inner product}{G inner
product}
We can define a Hermitian inner product in
$\mathcal{C}_G$ (restricted from one on $\CC G$)
via
\[ \langle f_1, f_2 \rangle_G \defeq \frac{1}{|G|}
\sum_{g \in G} \ol{f_1(g)} f_2(g) \]
The indicator functions $\indicator{O_i}$ are
pairwise orthogonal with respect to $\langle
\bullet, \bullet \rangle_G$ and moreover,
\[ \langle \indicator{O_i}, \indicator{O_i}
\rangle_G = \frac{1}{|G|} |O_i| =
\frac{1}{|C_G(x_i)|} \]
for any $x_i \in O_i$. Thus if $x_1, \ldots, x_r$
are representatives of $O_1, \ldots, O_r$, then
\[ \langle f_1, f_2 \rangle_G = \sum_{i = 1}^r
\frac{1}{|C_G(x_i)|} \ol{f_1(x_i)} f_2(x_i) \]
for $f_1, f_2 \in \mathcal{C}_G$.

\begin{example*}
  If $G = D_6 = \langle s, t \mid s^2 = t^3 = e,
  sts^{-1} = t^{-1} \rangle$ then
  \[ \langle f_1, f_2 \rangle_{D_6} = \frac{1}{6}
  \ol{f_1}(e) f_2(e) + \half \ol{f_1(s)} f_2(s) +
  \third \ol{f_1(t)} f_2(t) .\]
  In particular if $V$ is the natural \repdim{2}
  \gls{rep} of $D_6$ and $\CC$ is the
  \gls{triv_rep} then
  \[ \chichar_\CC = \indicator{G} \]
  \[ \chichar_V(e) = 2, \chichar_V(s) = 0,
  \chichar_V(t) = -1 \]
  \[ \langle \chichar_\CC, \chichar_\CC
  \rangle_{D_6} = \frac{1}{6} \cdot 1 + \half
  \cdot 1 + \third
  \cdot 1 = 1 \]
  \[ \langle \chichar_V, \chichar_V \rangle_{D_6}
  = \frac{1}{6} 2^2 + \half 0^2 + \third (-1)^2 =
 1 \]
 \[ \langle \chichar_\CC, \chichar_V \rangle_{D_6}
 = \frac{1}{6} 2 + \half 0 + \third (-1) = 0 \]
\end{example*}

\begin{lemma*}
  If $V$ and $W$ are (\gls{uni_rep}) \glspl{rep}
  of $G$ then
  \[ \chichar_{\Hom_k(V, W)}(g) =
  \ol{\chichar_V}(g) \chichar_W(g) \qquad \forall
  g \in G \]
\end{lemma*}

\begin{proof}
  Given $g \in G$ we can choose bases $v_1,
  \ldots, v_n$ of $V$ and $w_1, \ldots, w_m$ of
  $W$ such that $g \cdot v_i = \lambda_i v_i$ and $g
  \cdot w_j = \mu_j w_j$ for some $\lambda_1,
  \ldots, \lambda_n, \mu_1, \ldots, \mu_n \in
  \CC$. Then the functions $\alpha_{ij}(v_k) =
  \delta_{jk} w_i$ extend to linear maps
  $\alpha_{ij} \in \Hom_k(V, W)$ that form a basis
  (with respect to given basis $\alpha_{ij}$ as
  \glsref[rep]{represented} by a matrix with $0$s
  everywhere except a single $1$ in the $i, j$
  position).
  \begin{align*}
    (g \cdot \alpha_{ij})(v_k)
    &= g(\alpha_{ij} (g^{-1} v_k)) \\
    &= g(\alpha_{ij}(\lambda_k^{-1} v_k)) \\
    &= \lambda_k^{-1}(g(\delta_{jk} w_i)) \\
    &= \lambda_k^{-1} \mu_i \delta_{jk} w_i
  \end{align*}
  i.e. $g \cdot \alpha_{ij} = \lambda_j^{-1} \mu_i
  \alpha_{ij}$. So $\chichar_{\Hom_k(V, W)}(g) =
  \sum_{i, j} \lambda_j^{-1} \mu_i =
  \chichar_V(g^{-1}) \chichar_W(g) =
  \ol{\chichar_V(g)} \chichar_W(g)$.
\end{proof}

\begin{flashcard}[dim-of-Gfix-lemma]
\begin{lemma*}
  If $U$ is a \gls{rep} of $G$ then
  \[ \dim U^G = \cloze{\frac{1}{|G|} \sum_{g \in G}
  \chichar_V(g) = \langle \indicator{G},
  \chichar_V \rangle_G} .\]
\end{lemma*}

\begin{proof}
  \cloze{
  We've seen \hyperref[UG_projection]{before} that $\pi : U \to U_j$,
  $\pi(u) = \frac{1}{|G|} \sum_{g \in G} g \cdot
  u$ is a projection of $U$ onto $U^G$ and so
  $\dim U^G = \Trace \pi = \frac{1}{|G|} \sum_{g
  \in G} \Trace(g) = \frac{1}{|G|} \sum_{g \in G}
  \chichar_U(g)$.
  }
\end{proof}
\end{flashcard}

\begin{flashcard}[dim-HomG-V-W-prop]
\begin{proposition*}
  If $V$ and $W$ are any \glspl{rep} of $G$ then
  \[ \dim \GHom_G(V, W) = \cloze{\langle \chichar_V,
  \chichar_W \rangle_G} \]
\end{proposition*}

\begin{proof}
  \cloze{
  \begin{align*}
    \dim \GHom_G(V, W)
    &= \dim (\Hom_k(V, W))^G \\
    &= \langle \indicator{G}, \chichar_{\Hom_k(V, W)}
    \rangle_G \\
    &= \frac{1}{|G|} \sum_{g \in G}
    \ol{\chichar_V}(g) \chichar_W(g) \\
    &= \langle \chichar_V, \chichar_W \rangle_G
    \qedhere
  \end{align*}
  }
\end{proof}
\end{flashcard}

\begin{flashcard}[orthogonality-of-chars-thm]
\begin{theorem*}[Orthogonality of characters]
  \label{orthog_of_chars}
  If $V$ and $W$ are \gls{irred}
  $\CC$-\glspl{rep} of $G$, then
  \[ \langle \chichar_V, \chichar_W \rangle
  =\cloze{
  \begin{cases}
    1 & V \simeq W \\
    0 & V \not\simeq W
  \end{cases} } \]
\end{theorem*}

\begin{proof}
  \cloze{
  Use the fact that $\dim \GHom_G(V, W) = \langle
  \chichar_V, \chichar_W \rangle_G$ and
  \nameref{schurs_lemma}. If $\chichar_V =
  \chichar_W$ with $V$ and $W$ \gls{irred} then
  $\langle \chichar_V, \chichar_W \rangle_G =
  \langle \chichar_V, \chichar_V \rangle_G > 0$
  since $\chichar_V \neq 0$ so $\dim \GHom_G(V, W)
  > 0$ and $V \simeq W$ by \nameref{schurs_lemma}.
  }
\end{proof}
\end{flashcard}

\begin{flashcard}[rep-decomp-coro]
\begin{corollary*}
  If $(\rho, V)$ is a \gls{rep} of $G$ then
  \[ V \simeq \bigoplus_{\text{$W$ \gls{irred}
  \glspl{rep} of $G / \simeq$}} \langle
  \chichar_W, \chichar_\rho \rangle_G W \]
  In particular if $\sigma$ is a another \gls{rep}
  with $\chichar_\rho = \chichar_\sigma$ then
  $\sigma \simeq \rho$.
\end{corollary*}

\begin{proof}
  \cloze{
  By \nameref{maschkes_thm} there are $n_w \ge 0$
  such that
  \[ V \simeq \bigoplus_{\text{$W$ \gls{irred}}}
  n_w W \]
  Moreover, we've seen before that $n_w = \dim
  \GHom_G(W, V) = \langle \chichar_W,
  \chichar_\rho \rangle_G$ by the previous
  Proposition. So the first part follows.

  Since
  \[ \bigoplus_{\text{$W$ \gls{irred}}} \langle
  \chichar_W, \chichar_\rho \rangle_G W \]
  depends only on $\chichar_\rho$ (up to
  \gls{rep_isom}), the second part follows.
  }
\end{proof}
\end{flashcard}

Notice this proof depends on
\nameref{maschkes_thm} / \gls{comp_red} as well as
orthogonality of \glsref[char_rep]{characters}. For example, if we
have the two representations of $(\ZZ, +)$
determined by
\[ \rho(1) =
\begin{pmatrix}
  1 & 0 \\
  0 & 1
\end{pmatrix}
\qquad \sigma(1) =
\begin{pmatrix}
  1 & 1 \\
  0 & 1
\end{pmatrix}
\]
they are not \gls{rep_isom} but have the same
\glsref[char_rep]{characters}. $\rho(n) = \sigma(n) = 2
~\forall n \in \ZZ$.
Indeed both have trivial \glspl{subrep} $\langle
e_1 \rangle$ with trivial quotients. Slogan:
``\glsref[char_rep]{characters} cannot see gluing
data''.

\begin{flashcard}[irred-iff-chars-coro]
\begin{corollary*}
  If $\rho$ is a $\CC$-\gls{rep} of $G$ with
  \gls{char_rep} $\chichar$ then
  \[ \text{$\rho$ is \gls{irred}} \iff \cloze{\langle
  \chichar, \chichar \rangle_G = 1} .\]
\end{corollary*}

\begin{proof}
  \phantom{}
  \begin{enumerate}[$\Rightarrow$]
    \item[$\Rightarrow$] \cloze{Is clear from
      orthogonality of
      \glsref[char_rep]{characters}.}
    \item[$\Leftarrow$] \cloze{$\rho$ decomposes as $\rho
      \simeq \bigoplus n_w W$ with $n_w \ge 0$.
      Then $\chichar = \sum n_w \chichar_W$ but
      \[ \langle \chichar, \chichar \rangle_G =
      \sum n_w^2 \]
      so $\langle \chichar, \chichar \rangle_G =
      1 \implies \chichar = \chichar_w$ for some
      $W$.} \qedhere
  \end{enumerate}
\end{proof}
\end{flashcard}

This is a good way to prove
\glsref[irred]{irreducibility}.

\begin{example*}
  If $V$ is the natural \repdim{2} \gls{rep} of
  $D_6$ then $\langle \chichar_V, \chichar_V
  \rangle_{D_6} = 1$ and so $V$ is \gls{irred}.
\end{example*}

\begin{theorem*}[The character table is square]
  The \gls{irred} \glsref[char_rep]{characters} of
  $G$ form an orthonormal basis of $\mathcal{C}_G$
  with respect to $\langle \bullet, \bullet
  \rangle_G$.
\end{theorem*}

\begin{proof}
  We've already seen that the \gls{irred}
  \glsref[char_rep]{characters} form an
  orthonormal set so it remains to prove that they
  span. Let $I = \langle \chichar_1, \ldots,
  \chichar_s \rangle$ be their $\CC$-linear span.

  It suffices to show that
  \[ I^\perp \defeq \{g \in \mathcal{C}_G :
  \langle f, \chichar_i \rangle_G = 0 \text{ for
  $i = 1, \ldots, s$}\} = 0 .\]

  For $f \in \mathcal{C}_G$ and $(\rho, V)$ a
  \gls{rep} of $G$ show
  \[ \varphi_{f, V} = \frac{1}{|G|} \sum_{g \in G}
  \ol{f(g)} \rho(g) \in \GHom_G(V, V) \]
  and use \nameref{schurs_lemma} to show if $f \in
  I^\perp$ then $\varphi_{f, v} = 0$. Finally show
  $0 = \varphi_{f, \CC G} \delta_e = \frac{1}{|G|}
  \ol{f}$ so $f = 0$.
  \[ I = \langle \chichar_1, \ldots, \chichar_s
  \rangle \]
  where $\chichar_1,\ldots, \chichar_s$ are the
  \gls{irred} \glsref[char_rep]{characters}. For $f
  \in \mathcal{C}_G$ and a \gls{rep} $(\rho, V)$
  we can define
  \[ \varphi_{f, V} = \varphi = \frac{1}{|G|}
  \sum_{g \in G} \ol{f(g)} \rho(g) \in
  \GHom_\CC(V, V) \]
  If $h \in G$ then
  \begin{align*}
    \rho(h)^{-1} \varphi \rho(h)
    &= \frac{1}{|G|} \sum_{g \in G} \ol{f(g)}
    \rho(h^{-1} gh) \\
    &= \frac{1}{|G|} \sum_{g \in G}
    \ol{f(hgh^{-1})} \rho(g)
    &&\text{since $g \mapsto hgh^{-1}$ is a
    bijection} \\
    &= \frac{1}{|G|} \sum_{g \in G} \ol{f(g)}
    \rho(g)
    &&\text{since $f \in \mathcal{C}_G$} \\
    &= \varphi
  \end{align*}
  So $\varphi \rho(h) = \rho(h) \varphi ~\forall h
  \in G$ and $\varphi \in \GHom_G(V, V)$. If in
  particular, $(\rho, V)$ is \gls{irred}, then
  $\exists \lambda \in \CC$ such that $\varphi_{f,
  V} = \lambda \id_\CC$ since $\CC$ is
  algebraically closed. Then $\Gip\langle f,
  \chichar_\rho \rangle_G = \Trace \varphi_{f, V}
  = \lambda \dim V$. So if $f \in I^\perp$ then
  $\lambda = 0$ and $\varphi_{f, V} = 0$. But in
  general if $(\rho, V)$ is any \gls{rep}, then $V
  \simeq \bigoplus V_i$ for some \gls{irred}
  \glspl{rep} $V_i$ (\nameref{maschkes_thm}) and
  $\rho = \bigoplus \rho_i$ and $\varphi_{f, V} =
  \bigoplus \varphi_{f, V_i}$. So again if $f \in
  I^\perp$, then $\varphi_{f, V} = 0$. In
  particular if $V = \CC G$ is the \gls{reg_rep}
  then
  \[ 0 = \varphi_{f, \CC G} \delta_e =
  \frac{1}{|G|} \sum_{g \in G} \ol{f(g)} \delta_g
  = \frac{1}{|G|} \ol{f} \]
  So $f = 0$ and $I^\perp = 0$.
\end{proof}
