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\section{Complete Reducibility and Maschke's Theorem}

\textbf{Question:} What can a \gls{rep} $V$
of a group $G$ be decomposed as a direct sum of
\gls{simple} \glspl{subrep}?

\begin{example*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item If $G = \{e\}$ the answer is always as
      seen in \lecture{1} since a \gls{simple}
      \gls{subrep} is precisely a 1
      dimensional subspace.
    \item If $G = C_2$ and $V = \RR^2$,
      \[ \rho(-1) =
      \begin{pmatrix}
        -1 & 0 \\
        0 & 1
      \end{pmatrix}
      \]
      we've seen that the \gls{proper_subrep}
      \glspl{subrep} of $V$ are
      \[ \left\langle
      \begin{pmatrix}
        1 \\
        0
      \end{pmatrix}
      \right\rangle, \left\langle
      \begin{pmatrix}
        0 \\
        1
      \end{pmatrix}
      \right\rangle \]
      and
      \[ \RR^2 = \left\langle
      \begin{pmatrix}
        1 \\
        0
      \end{pmatrix}
      \right\rangle \oplus \left\langle
      \begin{pmatrix}
        0 \\
        1
      \end{pmatrix}
      \right\rangle  \]
      is the only such decomposition.
    \item If $G = (\ZZ, +)$ and $\rho : G \to
      \GL_2(k)$ is determined by
      \[ \rho(1) =
      \begin{pmatrix}
        1 & 1 \\
        0 & 1
      \end{pmatrix}
      \]
      then there is precisely one
      \gls{proper_subrep}
      \gls{subrep}
      \[ \left\langle
      \begin{pmatrix}
        1 \\
        0
      \end{pmatrix}
      \right\rangle \]
      as any such must be spanned by an
      eigenvector. So this cannot be decomposed in
      this way.
  \end{enumerate}
\end{example*}

\begin{flashcard}[direct-sum-reps-defn]
\begin{definition*}[direct sum of representations]
  \glsnoundefn{rep_dir_sum}{direct sum}{direct sums}
  \glssymboldefn{rep_dir_sum}{bigoplus}{direct sum
  of representations}
  \cloze{
  We say a \gls{rep} $V$ is a \emph{direct
  sum of $(V_i)_{i = 1}^k$} if each $V_i$ is a
  \gls{subrep} of $V$ and $V = \bigoplus_{i =
  1}^k V_i$ as vector spaces (recall direct sum
  notation from Linear Algebra). Given a family of
  \glspl{rep} $(\rho_i, V_i)_{i = 1}^k$ of
  $G$, we may define the (external) direct sum to
  be the \gls{rep} of $G$ on the vector space
  \[ V \defeq \bigoplus_{i = 1}^k V_i \defeq
  \{(v_i)_{i = 1}^k \mid v_i \in V_i\} \]
  with pointwise operations via
  \[ \rho(g)((v_i)) = (\rho_i(g) v_i) \]
  We write
  \[ (\rho, V)
  \defeq \bigoplus_{i = 1}^k (\rho_i, V_i)
  = \bigoplus_{i = 1}^k \rho_i
  = \bigoplus_{i = 1}^k V_i \]
  }
\end{definition*}
\end{flashcard}

\subsubsection*{Examples}

\begin{enumerate}[(1)]
  \item Suppose $G$ acts on a finite set $X$ and
    $X = X_1 \cup X_2$ with $X_1 \cap X_2 =
    \emptyset$ and $X_1$, $X_2$ both $G$ stable
    ($g \cdot x \in X_i$ if $x \in X_i$, $g \in
    G$). Then $kX \simeq k X_i \oplus k X_2$ under
    \[ f \mapsto (f|_{X_1}, f|_{X_2}) \]
    Internally
    \[ kX = \{f \mid f(x) = 0 ~\forall x \in X_2\}
    \oplus \{f \mid f(x) = 0 ~\forall x \in X_1\} \]
    More generally if the $G$-action decomposes
    into orbit $X = \bigcup_{i = 1}^r O_i$, then
    \[ kX = \bigoplus_{i = 1}^r \mathbbm{1}_{O_i}
    (kX) \simeq \bigoplus_{i = 1}^r k O_i \]
    where $\mathbbm{1}_{O_i} : kX \to kX$ given by
    \[ \mathbbm{1}_{O_i} (f)(x) = \begin{cases}
      f(x) & x \in O_i \\
      0 & \text{otherwise}
    \end{cases} \]
  \item If $G$ acts transitively on a finite set
    $X$ then
    \[ U \defeq \left\{ f \in kX ~\Bigg|~ \sum_{x
    \in X} f(x) = 0 \right\} \qquad \text{and}
    \qquad
    W \defeq \{ f \in kX \mid \text{$f$
    is constant}\} \]
    are \glspl{subrep} if $|X| > 1$.
    \begin{proof}
      If $f \in U$ and $g \in G$
      \[ \sum_{x \in X} (g \cdot f) (x) = \sum_{x
      \in X} f(g^{-1} x) = \sum_{x \in X} f(x) = 0 \]
      since $x \mapsto g^{-1} x$ defines a
      permutation of $X$. So $g \cdot f \in U$ and
      $U$ is \Ginv. Similarly if $f \in W$
      then there exists $\lambda \in k$ such that
      for all $x \in X$, $f(x) = \lambda$ and $(g
      \cdot f) (x) = f(g^{-1} x) = \lambda$. If
      $\characteristic k = 0$ then $kX = U \oplus W$ is a
      \gls{rep_dir_sum} of \glspl{rep}. What happens
      if $\characteristic k = p > 0$?
    \end{proof}
\end{enumerate}

\begin{proposition*}
  Suppose $\rho : G \to \GL(V)$ is a
  \gls{rep} and $V = U \oplus W$ as
  vector spaces. Then the following are
  equivalent:
  \begin{enumerate}[(i)]
    \item $V = U \oplus W$ as
      \glspl{rep}
    \item There is a basis $v_1, \ldots,
      v_d$ of $V$ such that $v_1, \ldots,
      v_r$ is a basis of $U$ and $v_{r + 1},
      \ldots, v_d$ is a basis of $W$ and the
      corresponding matrices $\rho(g)$ are
      block diagonal
      \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/acfc851869b411ee.png}
      \end{center}
    \item For every basis $v_1, \ldots, v_d$
      of $V$ such that $v_1, \ldots, v_r$ is
      a basis of $U$ and $v_{r + 1}, \ldots,
      v_d$ is a basis of $W$, the
      corresponding matrices $\rho(g)$ are
      all block diagonal as in (ii).
  \end{enumerate}
\end{proposition*}

\begin{proof}
  Think about it!
\end{proof}

\begin{warning*}
  $\rho : C_2 \to \GL_2(\RR)$, $\rho(-1) =
  \begin{pmatrix} -1 & 2 \\ 0 & 1
  \end{pmatrix}$ defines a \gls{rep} of
  $C_2$ (check). The \gls{rep} on
  $\RR^2$ decomposes as a \gls{rep_dir_sum} of
  \glspl{subrep} $\langle e_1 \rangle$
  and $\langle e_1 + e_2 \rangle$ even
  though $\rho(-1)$ is not diagonal.
\end{warning*}

\begin{flashcard}[completely-reducible-defn]
\begin{definition*}[completely reducible]
  \glsdefn{comp_red}{completely reducible}{asdf}
  \cloze{We say a \gls{rep} $V$ of a group $G$ is
  \emph{completely reducible} if
  \[ V \simeq \bigoplus_{i = 1}^r V_i \]
  for some \gls{irred} \gls{rep} $V_i$.}
\end{definition*}
\end{flashcard}
\vspace{-1em}

We've seen that $(\ZZ, +)$ has \glspl{rep}
that are not \gls{comp_red}, given by
\[ \rho(1) =
\begin{pmatrix}
  1 & 1 \\
  0 & 1
\end{pmatrix}
\]

\begin{flashcard}[completely-reducible-lemma]
\begin{lemma*}
  Suppose $(\rho, V)$ is a \gls{rep} such
  \cloze{that for every pair of \Ginv{}
  subspaces $W_1, W_2 \le V$ such that $W_1 \le
  W_2$, there is a \Ginv\ complement to
  $W_1$ in $W_2$.}

  Then $V$ is \gls{comp_red}.
\end{lemma*}

\begin{proof}
  \cloze{
  By induction on $\dim V$. If $V = 0$ or $V$ is
  \gls{irred} the result is clear. Otherwise $V$
  has a proper \Ginv\ subspace $W$. Then by
  assumption $W$ has a \Ginv\ complement
  $U$ in $V$ so $V = U \oplus W$ as
  \glspl{rep}.

  Now $\dim U, \dim W < \dim V$ and $U$ and $W$
  inherit the condition on $V$. So by induction
  hypothesis,
  \[ U \cong \bigoplus_{i = 1}^r U_i \qquad
  \text{and} \qquad W \cong \bigoplus_{j = 1}^s
  W_j \]
  for some \gls{simple} \glspl{rep} $U_1, \ldots,
  U_r$ and $W_1, \ldots, W_s$. Then
  \[ V \simeq \bigoplus_{i = 1}^r U_i \oplus
  \bigoplus_{j = 1}^s W_j \]
  as required.
  }
\end{proof}
\end{flashcard}

Recall, if $V$ is a $\CC$-vector space then a
\emph{Hermitian inner product} is a positive
definite Hermitian sesquilinear form, i.e.
$(\bullet, \bullet) : V \times V \to \CC$ such
that
\begin{enumerate}[(i)]
  \item Sesquilinear:
    \[ (ax + by, z)
    = \ol{a}(x, z) + \ol{b}(y, z)
    \qquad \forall x, y, z \in V, a, b \in \CC \]
    \[ (x, ay + bz)
    = a(x, y) + b(x, z)
    \qquad \forall x, y, z \in V, a, b \in \CC \]
  \item $(x, y) = \ol{(y, x)}$ for all $x, y \in
    V$ (Hermitian)
  \item $(x, x) > 0 ~\forall x \in V \setminus
    \{0\}$ (positive definite)
\end{enumerate}
The standard inner product on $\CC^n$ is given by
\[ (x, y) = \sum_{i = 1}^n \ol{x_i} y_i \]
Recall the \emph{unitary group} $U(n)$ is the
subgroup of $\GL_n(\CC)$ given by
\begin{align*}
  U(n)
  &= \{A \in \GL_n(\CC) \mid \ol{A^\top} A = I\}
  \\
  &= \{A \in \GL_n(\CC) \mid (Ax, Ay) = (x, y)
  ~\forall x, y \in \CC^n\}
\end{align*}

\begin{flashcard}[unitary-rep-defn]
\begin{definition*}[unitary representation]
  \glsadjdefn{uni_rep}{unitary}{\gls{rep}}
  \cloze{
  A $\CC$ \gls{rep} of a group $G$ is
  \emph{unitary} if there exists a basis $e_1,
  \ldots, e_n$ of $V$ so that the corresponding
  matrix \gls{rep} $\rho : G \to \GL_n(\CC)$
  has image contained in $U(n)$.
  }
\end{definition*}
\end{flashcard}
