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\section{Introduction}

What is Representation Theory?

\newcommand{\undertext}[2] {$\underbrace{\textrm{#1}}_{\textrm{#2}}$}
The study of how \undertext{symmetry}{groups}
\undertext{occurs}{act linearly on} in
\undertext{nature}{finite dimensional vector spaces}.

\textbf{Main goal:} Understand for a given group
$G$ all the ways it can act linearly on a finite
dimensional vector space, i.e. classify them.
Subproblem: What does it mean for two such to be
the ``same''? How do they break into smaller
pieces?

\textbf{Secondary goal:} Use representations to
understand groups, e.g. give a proof that no
finite simple group has order with precisely two
prime factors.

\subsection{Linear Algebra Revision}

By vector space we will always mean finite
dimensional vector space (unless we say not) over
a field $k$. $k$ will usually be algebraically
closed and characteristic zero, for example $\CC$,
but that is because it is an easy first case, but
theories are normally more general and sometimes
we'll look at these.

Given a vector space $V$ we define the general
linear group of $V$
\[ \GL(V) = \Aut(V) = \{\alpha : V \hookrightarrow V
\mid \text{$\alpha$ is $k$-linear and invertible}\} \]
This is a group under composition of linear maps.

Since $V$ is finite dimensional, there is a
(linear) isomorphism $k^d \simeq V$ for some $d
\ge 0$ called the dimension. The choice of
isomorphism determines a basis $e_1, \ldots, e_d$
of $B$ where $e_1$ is the image of the $i$-th
standard basis vector under the isomorphism.

Then
\[ \GL(B) \simeq \ub{\{A \in \Mat_d(k) \mid \det A
\neq 0\}}_{\text{group under matrix multiplication}} \]
This isomorphism sends a linear map $\alpha$ to
the matrix $A_{ij}$ such that
\[ \alpha(e_i) = \sum_{j = 1}^d A_{ji} e_j \]

\textbf{Exercise:} check that this does define an
isomorphism of groups.

The choice of isomorphism also gives a
decomposition of $V$ as a direct sum of $1$
dimensional subspaces
\[ V = \bigoplus_{i = 1}^d ke_i \]
This decomposition is not unique unless $d = 1$,
but the number of summands is always $\dim V$.

\subsection{Group Representation -- Definitions and Examples}

Recall that an action of a group $G$ on a set $X$
is a function $\cdot : G \times X \to X$, $(g, x)
\mapsto g \cdot x$ such that
\begin{enumerate}[(i)]
  \item $e \cdot x = x ~\forall x \in X$;
  \item $g \cdot (h \cdot x) = (gh) \cdot x
    ~\forall g, h \in G, x \in X$.
\end{enumerate}
Recall also that to define an action is equivalent
to defining a group homomorphism $\rho : G \to
S(X)$ where $S(X)$ is the symmetric group of $X$
i.e. the set of bijections $X \to X$ with operation
composition of functions via $\rho(g)(x) = g \cdot
x$ for all $g \in G, x \in X$.

\begin{flashcard}[representation-defn]
\begin{definition*}[Representation]
  \glsnoundefn{rep}{representation}{representations}
  \cloze{
  A \emph{representation} of a group $G$ on a
  vector space $V$ is a group homomorphism $\rho :
  G \to \GL(V)$.
  }
\end{definition*}
\end{flashcard}

\begin{notation*}
  By abuse of notation, we'll sometimes call the
  \gls{rep} $\rho$, sometimes $(\rho, V)$ and
  sometimes just $V$.
\end{notation*}
\vspace{-1em}

Defining a \gls{rep} of $G$ in $V$
corresponds to assigning a linear map $\rho(g) : V
\to V$ to each $g \in G$ such that
\begin{enumerate}[(i)]
  \item $\rho(e) = \id_V$
  \item $\rho(gh) = \rho(g)\rho(h)$
  \item $\rho(g^{-1}) = \rho(g)^{-1}$
\end{enumerate}
\textbf{Exercise:} Show that if condition (ii)
holds then (i) is equivalent to (iii). Moreover,
both can be replaced by $\rho(g) \in \GL(V)
~\forall g \in G$.

Given a basis for $V$ a \gls{rep} can be
viewed as an assignment of matrix $\rho(g)$ in
$\Mat_{\dim V}(k)$ to each $g \in G$ such that
(i), (ii) and (iii) hold.

\begin{flashcard}[degree-of-representation-defn]
\begin{definition*}[Degree of representation]
  \glspropdefn{rep_deg}{degree}{\gls{rep}}
  \glspropdefn[rep_deg]{rep_dim}{dimension}{\gls{rep}}
  \cloze{
  The \emph{degree} of $\rho$ or \emph{dimension} of
  $\rho$ is $\dim V$.
  }
\end{definition*}
\end{flashcard}

\begin{flashcard}[faithful-representation-defn]
\begin{definition*}[Faithful representation]
  \glspropdefn{ff_rep}{faithful}{\gls{rep}}
  \cloze{
  $\rho$ is \emph{faithful} if $\ker \rho =
  \{e\}$.
  }
\end{definition*}
\end{flashcard}

\subsubsection*{Examples}

\begin{enumerate}[(1)]
  \item \glsnoundefn{triv_rep}{trivial
    representation}{trivial representations}
    Let $G$ be any group and $V = k$. Then
    $\rho : G \to \GL(k)$, $g \mapsto \id$ is
    called the \emph{trivial representation}.
  \item Let $G = C_2 = (\{\pm 1\}, \cdot)$, $V =
    \RR^2$ then
    \[ \rho(1) = \begin{pmatrix}
      1 & 0 \\
      0 & 1
    \end{pmatrix} \]
    \[ \rho(-1) = \begin{pmatrix}
      -1 & 0 \\
      0 & 1
    \end{pmatrix} \]
    defines a \gls{rep} of $G$ on $V$ since
    $\rho(-1)^2 = \rho(1)$.
  \item Let $G = (\ZZ, +)$, $V$ a vector space and
    $\rho$ a \gls{rep} of $G$ on $V$.
    Necessarily $\rho(0) = \id_V$, $\rho(1) : V
    \hookrightarrow V$ is an invertible linear map
    $\alpha$, say $\rho(1 + 1) = \rho(1) =
    \alpha^2$. By induction $\rho(n) = \alpha^n$
    for all $n \ge 0$, and for $n < 0$, $\rho(n) =
    \rho(-n)^{-1} = (\alpha^{-n})^{-1} = \alpha^n$
    so $\rho(n) = \alpha^n$ for all $n \in \ZZ$.

    Notice conversely for any $\alpha \in \GL(V)$,
    $\rho(n) = \alpha^n ~\forall n \in \ZZ$
    defines a \gls{rep} of $G$ on $V$. So
    \begin{align*}
      \{\text{\glspl{rep} of $G$ on $V$}\}
      &\stackrel{1-1}{\leftrightarrow} \GL(V) \\
      \rho
      &\mapsto \rho(1)
    \end{align*}
  \item Let $G = (\ZZ / N\ZZ, +)$ and $\rho : G
    \to \GL(V)$ a \gls{rep}. As before
    $\rho(n + N\ZZ) = \rho(1 + N\ZZ)^n$ for all $n
    \in \ZZ$. But now $\rho(N + N\ZZ) = \rho(0 +
    N\ZZ) = \id_V$ so $\rho(1 + N\ZZ)^N = \id_V$.
    So
    \begin{align*}
      \{\text{\glspl{rep} of $(\ZZ/N\ZZ, +)$ on $V$}\}
      &\stackrel{1-1}{\leftrightarrow} \{\alpha
      \in \GL(V) \mid \alpha^N = \id\} \\
      \rho
      &\mapsto \rho(1 + N\ZZ)
    \end{align*}
  \item $G = S_3 = S(\{1, 2, 3\})$ and $V =
    \RR^2$. Take an equilateral triangle in
    $\RR^2$ centred at the origin and labelled
    vertices $1, 2, 3$.
    \begin{center}
      \includegraphics[width=0.6\linewidth]
      {images/a646defe643611ee.png}
    \end{center}
    Then $G$ acts on the triangle by permuting
    vertices. Each such symmetry induces a linear
    transformation of $\RR^2 = V$. For example $g
    = (12)$ induces reflection in the line through
    the origin and 3, and $g = (123)$ induces a
    rotation by $\frac{2\pi}{3}$.

    \textbf{Exercise:} Choose a basis for $\RR^2$.
    Write down the coordinates of the vertices of
    triangle. For $g \in S_3$ write down the
    matrix of the induced linear map. Check it
    defines a \gls{rep}. Would a different
    basis have made the calculation easier?
\end{enumerate}