\chapter{Low-Dimensional Representations}\label{ch:representations}

Having built a solid understanding of the combinatorics of Dehn twists, we are
now ready to attack the problem of classifying the representations of
\(\Mod(\Sigma_g)\) of sufficiently small dimension. As promised, our strategy
is to make use of the \emph{geometrically-motivated} relations derived in
Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations}.

Historically, these relations have been exploited by Funar \cite{funar},
Franks-Handel \cite{franks-handel} and others to establish the triviality of
low-dimensional representations, culminating in Korkmaz' \cite{korkmaz} recent
classification of representations of dimension \(n \le 2 g\) for \(g \ge 3\).
The goal of this chapter is to provide a concise account of Korkmaz' results.

\begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial}
  Let \(\Sigma_g^b\) be the compact surface of genus \(g \ge 1\) with \(b\)
  boundary components and \(\rho : \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\) be
  a linear representation with \(n < 2 g\). Then the image of \(\rho\) is
  Abelian. In particular, if \(g \ge 3\) then \(\rho\) is trivial.
\end{theorem}

Like some of the results we have encountered so far, the proof of
Theorem~\ref{thm:low-dim-reps-are-trivial} is elementary in nature: we proceed
by induction on \(g\) and tedious case analysis. We begin by the base case \(g
= 2\).

\begin{proposition}\label{thm:low-dim-reps-are-trivial-base-case}
  Given \(\rho : \Mod(\Sigma_2^b) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the
  image of \(\rho\) is Abelian.
\end{proposition}

\begin{proof}[Sketch of proof]
  Given \(\alpha \subset \Sigma_2^b\), let \(L_\alpha = \rho(\tau_\alpha)\) and
  denote by \(E_{\alpha = \lambda} = \{ v \in \mathbb{C}^n : L_\alpha v =
  \lambda v \}\) its eigenspaces. Let \(\alpha_1, \alpha_2, \beta_1, \beta_2,
  \gamma, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_2^b\) be the curves of the
  Lickorish generators from Theorem~\ref{thm:lickorish-gens}, as shown in
  Figure~\ref{fig:lickorish-gens-genus-2}.
  \begin{figure}
    \centering
    \includegraphics[width=.2\linewidth]{images/lickorish-gens-gen-2.eps}
    \caption{The Lickorish generators for $g = 2$.}
    \label{fig:lickorish-gens-genus-2}
  \end{figure}

  If \(n = 1\) then \(\rho(\Mod(\Sigma_2^b)) \subset \GL_1(\mathbb{C}) =
  \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by
  Proposition~\ref{thm:commutator-normal-gen} it suffices to show \(L_{\alpha_1}
  = L_{\beta_1}\), so that \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker
  \rho\) and thus \(\Mod(\Sigma_2^b)' \subset \ker \rho\) -- i.e.
  \(\rho(\Mod(\Sigma_2^b))\) is Abelian. Given the braid relation
  \begin{equation}\label{eq:braid-rel-induction-basis}
    L_{\alpha_1} L_{\beta_1} L_{\alpha_1}
    = L_{\beta_1} L_{\alpha_1} L_{\beta_1},
  \end{equation}
  this amounts to showing \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute.

  To that end, we exhaustively analyze all of the possible Jordan forms
  \begin{align*}
    \begin{pmatrix}
      \lambda & 0 \\
      0       & \lambda
    \end{pmatrix}
    & \quad{\normalfont(1)}
    &
    \begin{pmatrix}
      \lambda & 0 \\
      0       & \mu
    \end{pmatrix}
    & \quad{\normalfont(2)}
    &
    \begin{pmatrix}
      \lambda & 1 \\
      0       & \lambda
    \end{pmatrix}
    & \quad{\normalfont(3)}
    \\
    \begin{pmatrix}
      \lambda & 0       & 0       \\
      0       & \lambda & 0       \\
      0       & 0       & \lambda
    \end{pmatrix}
    & \quad{\normalfont(4)}
    &
    \begin{pmatrix}
      \lambda & 0   & 0   \\
      0       & \mu & 0   \\
      0       & 0   & \nu
    \end{pmatrix}
    & \quad{\normalfont(5)}
    &
    \begin{pmatrix}
      \lambda & 1       & 0       \\
      0       & \lambda & 1       \\
      0       & 0       & \lambda
    \end{pmatrix}
    & \quad{\normalfont(6)}
    \\
    \begin{pmatrix}
      \lambda & 0   & 0   \\
      0       & \mu & 1   \\
      0       & 0   & \mu
    \end{pmatrix}
    & \quad{\normalfont(7)}
    &
    \begin{pmatrix}
      \lambda & 0       & 0       \\
      0       & \lambda & 1       \\
      0       & 0       & \lambda
    \end{pmatrix}
    & \quad{\normalfont(8)}
    &
    \begin{pmatrix}
      \lambda & 0       & 0   \\
      0       & \lambda & 0   \\
      0       & 0       & \mu
    \end{pmatrix}
    & \quad{\normalfont(9)}
  \end{align*}
  of \(L_{\alpha_2}\) -- where \(\lambda, \mu, \nu \in \mathbb{C}^\times\) are
  all distinct. By changing basis we may assume without loss of generality that
  the matrix \(L_{\alpha_2}\) is exactly its Jordan form, so that \(E_{\alpha_2
  = \lambda} = \bigoplus_{i \le \dim E_{\alpha_2}} \mathbb{C} e_i\).

  For cases (1) to (6), we use the change of coordinates principle and
  different relations to show \(L_{\alpha_1}\) and \(L_{\beta_1}\) lie inside
  some Abelian subgroup of \(\GL_n(\mathbb{C})\).

  \begin{enumerate}[leftmargin=1.9cm]
    \item[\bfseries\color{highlight}(1) \& (4)]
      By the change of coordinates principle, both \(L_{\alpha_1}\) and
      \(L_{\beta_1}\) are conjugate to \(L_{\alpha_2} = \lambda\). But the
      only matrix conjugate to \(\lambda\) is \(\lambda\) itself. Hence
      \(L_{\alpha_1} = L_{\beta_1} = \lambda \in \mathbb{C}^\times\).

    \item[\bfseries\color{highlight}(2) \& (5)]
      Since \(\alpha_2\) is disjoint from both \(\alpha_1\) and \(\beta_1\), it
      follows from the disjointness relations \([\tau_{\alpha_1},
      \tau_{\alpha_2}] = [\tau_{\beta_1}, \tau_{\alpha_2}] = 1\) that
      \(L_{\alpha_1}\) and \(L_{\beta_1}\) preserve the eigenspaces of
      \(L_{\alpha_2}\), which are all \(1\)-dimensional. Hence \(L_{\alpha_1}\)
      and \(L_{\beta_1}\) lie inside the subgroup of diagonal matrices -- an
      Abelian subgroup of \(\GL_n(\mathbb{C})\).

    \item[\bfseries\color{highlight}(3) \& (6)]
      As before, it follows from the disjointness relations that \(E_{\alpha_2
      = \lambda} = \ker (L_{\alpha_2} - \lambda)\) and \(\ker (L_{\alpha_2} -
      \lambda)^2\) are invariant under both \(L_{\alpha_1}\) and
      \(L_{\beta_1}\). This implies \(L_{\alpha_1}\) and \(L_{\beta_1}\) are
      upper triangular matrices with \(\lambda\) along their diagonals. Any
      such pair of matrices satisfying the braid relation
      (\ref{eq:braid-rel-induction-basis}) commute.
  \end{enumerate}

  Similarly, in case (7) we use the braid relation and the disjointness
  relations to show \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute -- see
  \cite[Proposition~5.1]{korkmaz} for a full proof. Cases (8) and (9) require
  some extra thought. We consider the curve \(\beta_2\). In these cases, the
  eigenspace \(E_{\alpha_2 = \lambda}\) is \(2\)-dimensional. Since
  \(L_{\alpha_2}\) and \(L_{\beta_2}\) are conjugate, \(E_{\beta_2 = \lambda}\)
  is also \(2\)-dimensional -- indeed, conjugate operators have the same Jordan
  form. Now either \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) or
  \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\). We begin by the first
  case.

  We claim that if \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) then
  \(E_{\alpha_2 = \lambda}\) is \(\Mod(\Sigma_2^b)\)-invariant. Indeed, by
  Observation~\ref{ex:change-of-coordinates-crossing} we can always find \(f,
  g, h_i \in \Mod(\Sigma_2^b)\) with
  \begin{align*}
    f \cdot [\alpha_2]      & = [\alpha_1]
    &
    g \cdot [\alpha_2]      & = [\beta_1]
    &
    h_i \cdot [\alpha_2]    & = [\alpha_2]    \\
    f \cdot [\beta_2]   & = [\beta_1]
    &
    g \cdot [\beta_2]   & = [\gamma]
    &
    h_i \cdot [\beta_2] & = [\eta_i].
  \end{align*}
  In particular,
  \begin{align*}
    f   \tau_{\alpha_2}    f^{-1}   & = \tau_{\alpha_1}
    &
    g   \tau_{\alpha_2}    g^{-1}   & = \tau_{\beta_1}
    &
    h_i \tau_{\alpha_2}    h_i^{-1} & = \tau_{\alpha_2}     \\
    f   \tau_{\beta_2} f^{-1}   & = \tau_{\beta_1}
    &
    g   \tau_{\beta_2} g^{-1}   & = \tau_{\gamma}
    &
    h_i \tau_{\beta_2} h_i^{-1} & = \tau_{\eta_i}.
  \end{align*}
  and thus
  \begin{align*}
    E_{\alpha_1 = \lambda}
    = \rho(f) E_{\alpha_2 = \lambda}
    & = \rho(f) E_{\beta_2 = \lambda}
    = E_{\beta_1 = \lambda}
    \\
    E_{\beta_1 = \lambda}
    = \rho(g) E_{\alpha_2 = \lambda}
    & = \rho(g) E_{\beta_2 = \lambda}
    = E_{\gamma = \lambda}
    \\
    E_{\eta_i = \lambda}
    = \rho(h_i) E_{\alpha_2 = \lambda}
    & = \rho(h_i) E_{\beta_2 = \lambda}
    = E_{\beta_2 = \lambda}.
  \end{align*}
  In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} =
  E_{\beta_1 = \lambda} = E_{\beta_2 = \lambda} = E_{\gamma = \lambda} =
  E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1} = \lambda}\) is invariant
  under the action of all Lickorish generators.

  Hence \(\rho\) restricts to a subrepresentation \(\bar \rho :
  \Mod(\Sigma_2^b) \to \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) --
  recall \(E_{\alpha_2 = \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By
  case (2), \(\bar \rho(f) = 1\) for all \(f \in \Mod(\Sigma_2^b)'\), given
  that \(\bar \rho(\Mod(\Sigma_2^b))\) is Abelian. Thus
  \[
    \rho(\Mod(\Sigma_2^b)') \subset
    \begin{pmatrix}
      1 & 0 & * \\
      0 & 1 & * \\
      0 & 0 & *
    \end{pmatrix}
  \]
  lies inside the group of upper triangular matrices, a solvable subgroup of
  \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we
  get \(\rho(\Mod(\Sigma_2^b)') = 1\): any homomorphism from a perfect group to
  a solvable group is trivial.

  Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and
  the Jordan form of \(L_{\alpha_2}\) is given by (8) then the disjointness
  relations \([\tau_{\alpha_2}, \tau_{\alpha_1}] = [\tau_{\alpha_2},
  \tau_{\beta_1}] = [\tau_{\beta_2}, \tau_{\alpha_1}] = [\tau_{\beta_2},
  \tau_{\beta_1}] = 1\) implies that \(L_{\alpha_1}\) and \(L_{\beta_1}\)
  preserve the eigenspaces of both \(L_{\alpha_2}\) and \(L_{\beta_2}\), so
  \[
    0
    \subsetneq E_{\alpha_2 = \lambda} \cap E_{\beta_2 = \lambda}
    \subsetneq E_{\alpha_2 = \lambda}
    \subsetneq V
  \]
  is a flag of subspaces invariant under \(L_{\alpha_1}\) and \(L_{\beta_1}\).
  In this case we can find a basis for \(\mathbb{C}^3\) with respect to which
  the matrices of \(L_{\alpha_1}\) and \(L_{\beta_1}\) are both upper
  triangular with \(\lambda\) along the diagonal: take \(v_1, v_2, v_3 \in
  \mathbb{C}^3\) with \(v_1 \in E_{\alpha_2 = \lambda} \cap E_{\beta_2 =
  \lambda}\) and \(v_2 \in V_{L_{\alpha_2}}\). Any such pair of matrices
  satisfying the braid relation (\ref{eq:braid-rel-induction-basis}) commute.

  Similarly, if \(L_{\alpha_2}\) has Jordan form (9) and \(E_{\alpha_2 =
  \lambda} \ne E_{\beta_2 = \lambda}\) we use
  (\ref{eq:braid-rel-induction-basis}) to conclude \(L_{\alpha_1}\) and
  \(L_{\beta_1}\) commute -- again, see \cite[Proposition~5.1]{korkmaz}. We are
  done.
\end{proof}

We are now ready to establish the triviality of low-dimensional
representations.

\begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}]
  Let \(g \ge 1\), \(b \ge 0\), and fix \(\rho : \Mod(\Sigma_g^b) \to
  \GL_n(\mathbb{C})\) with \(n < 2g\). We want to show
  \(\rho(\Mod(\Sigma_g^b))\) is Abelian. As promised, we proceed by induction
  on \(g\). The base case \(g = 1\) is again clear from the fact \(n = 1\) and
  \(\GL_1(\mathbb{C}) = \mathbb{C}^\times\). The case \(g = 2\) was also
  established in Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}.

  Now suppose \(g \ge 3\) and every \(m\)-dimensional representation of
  \(\Sigma_{g - 1}^{b'}\) has Abelian image for \(m < 2(g - 1)\).
  Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots,
  \gamma_{g - 1}, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_g^b\) be the curves
  from the Lickorish generators of \(\Mod(\Sigma_g^b)\), as in
  Figure~\ref{fig:lickorish-gens}. Once again, let \(L_\alpha =
  \rho(\tau_\alpha)\) and denote by \(E_{\alpha = \lambda}\) the eigenspace of
  \(L_\alpha\) associated to \(\lambda \in \mathbb{C}\). Let \(\Sigma \cong
  \Sigma_{g - 1}^1\) be the closed subsurface highlighted in
  Figure~\ref{fig:korkmaz-proof-subsurface}.

  \begin{figure}[ht]
    \centering
    \includegraphics[width=.35\linewidth]{images/lickorish-gens-korkmaz-proof.eps}
    \caption{The subsurface $\Sigma \subset \Sigma_g^b$.}
    \label{fig:korkmaz-proof-subsurface}
  \end{figure}

  We claim that it suffices to find a \(m\)-dimensional
  \(\Mod(\Sigma)\)-invariant\footnote{Here we view $\Mod(\Sigma)$ as a subgroup
  of $\Mod(\Sigma_g^b)$ via the inclusion homomorphism $\Mod(\Sigma) \to
  \Mod(\Sigma_g^b)$ from Example~\ref{ex:inclusion-morphism}, which can be
  shown to be injective in this particular case.} subspace \(W \subset
  \mathbb{C}^n\) with \(2 \le m \le n - 2\). Indeed, in this case \(m < 2(g -
  1)\) and \(\dim \mfrac{\mathbb{C}^n}{W} = n - m < 2(g - 1)\). Thus both
  representations
  \begin{align*}
    \rho_1 : \Mod(\Sigma) & \to \GL(W) \cong \GL_m(\mathbb{C})
    &
    \rho_2 : \Mod(\Sigma) & \to \GL(\mfrac{\mathbb{C}^n}{W})
    \cong \GL_{n - m}(\mathbb{C})
  \end{align*}
  fall into the induction hypothesis -- i.e. \(\rho_i(\Mod(\Sigma))\) is
  Abelian. In particular, \(\rho_i(\Mod(\Sigma)') = 1\) and we can find some
  basis for \(\mathbb{C}^n\) with respect to which
  \[
    \rho(f) =
    \left(
    \begin{array}{c|c}
      1_m & *         \\ \hline
        0 & 1_{n - m}
    \end{array}
    \right)
  \]
  for all \(f \in \Mod(\Sigma)'\) -- where \(1_k\) denotes the \(k \times k\)
  identity matrix. Since the group of upper triangular matrices is solvable, it
  follows from Proposition~\ref{thm:commutator-is-perfect} that \(\rho\)
  annihilates all of \(\Mod(\Sigma)'\) and, in particular, \(\tau_{\alpha_1}
  \tau_{\beta_1}^{-1} \in \ker \rho\). Now recall from
  Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(\Sigma_g^b)'\) is
  normally generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we
  conclude \(\rho(\Mod(\Sigma_g^b)') = 1\), as desired.

  As before, we exhaustively analyze all possible Jordan forms of
  \(L_{\alpha_g}\). First, consider the case where we can find eigenvalues
  \(\lambda_1, \ldots, \lambda_k\) of \(L_{\alpha_g}\) such that the sum \(W =
  \bigoplus_i E_{\alpha_g = \lambda_i}\) of the corresponding eigenspaces has
  dimension \(m\) with \(2 \le m \le n - 2\). In this case, it suffices to
  observe that since \(\alpha_g\) lies outside of \(\Sigma\), each
  \(E_{\alpha_g = \lambda_i}\) is \(\Mod(\Sigma)\)-invariant: the Lickorish
  generators \(\tau_{\alpha_1}, \ldots, \tau_{\alpha_{g - 1}}, \tau_{\beta_1},
  \ldots, \tau_{\beta_{g - 1}}\), \(\tau_{\gamma_1}, \ldots, \tau_{\gamma_{g -
  2}}\) of \(\Sigma \cong \Sigma_{g - 1}^1\) all commute with
  \(\tau_{\alpha_g}\) and thus preserve the eigenspaces of its action on
  \(\mathbb{C}^n\).

  If no sum of the form \(\bigoplus_i E_{\alpha_g = \lambda_i}\) has dimension
  lying between \(2\) and \(n - 2\), then there must be at most \(2\) distinct
  eigenvalues \(\lambda\) of \(L_{\alpha_g}\), and \(\dim E_{\alpha_g =
  \lambda} = 1, n - 1, n\) for all such \(\lambda\). Hence the Jordan form of
  \(L_{\alpha_g}\) has to be one of
  \begin{align*}
    \begin{pmatrix}
      \lambda & 0       & 0      & \cdots & 0       & 0       \\
      0       & \lambda & 0      & \ldots & 0       & 0       \\
      \vdots  & \vdots  & \vdots & \ddots & \vdots  & \vdots  \\
      0       & 0       & 0      & \cdots & \lambda & 0       \\
      0       & 0       & 0      & \cdots & 0       & \lambda
    \end{pmatrix}
    & \quad{\normalfont(1)}
    &
    \begin{pmatrix}
      \lambda & 1       & 0      & \cdots & 0       & 0       \\
      0       & \lambda & 1      & \ldots & 0       & 0       \\
      \vdots  & \vdots  & \vdots & \ddots & \vdots  & \vdots  \\
      0       & 0       & 0      & \cdots & \lambda & 1       \\
      0       & 0       & 0      & \cdots & 0       & \lambda
    \end{pmatrix}
    & \quad{\normalfont(2)}
    \\
    \begin{pmatrix}
      \lambda & 0       & 0      & \cdots & 0       & 0       \\
      0       & \lambda & 0      & \ldots & 0       & 0       \\
      \vdots  & \vdots  & \vdots & \ddots & \vdots  & \vdots  \\
      0       & 0       & 0      & \cdots & \lambda & 1       \\
      0       & 0       & 0      & \cdots & 0       & \lambda
    \end{pmatrix}
    & \quad{\normalfont(3)}
    &
    \begin{pmatrix}
      \lambda & 0       & 0      & \cdots & 0       & 0       \\
      0       & \lambda & 0      & \ldots & 0       & 0       \\
      \vdots  & \vdots  & \vdots & \ddots & \vdots  & \vdots  \\
      0       & 0       & 0      & \cdots & \lambda & 0       \\
      0       & 0       & 0      & \cdots & 0       & \mu
    \end{pmatrix}
    & \quad{\normalfont(4)}
  \end{align*}
  for \(\lambda \ne \mu\). We analyze the first two sporadic cases
  individually.

  \begin{enumerate}
    \item[\bfseries\color{highlight}(1)]
      Here we use the change of coordinates principle: each \(L_{\alpha_i},
      L_{\beta_i}, L_{\gamma_i},  L_{\eta_i}\) is conjugate to \(L_{\alpha_g} =
      \lambda\), so all Lickorish generators of \(\Mod(\Sigma_g^b)\) act on
      \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence
      \(\rho(\Mod(\Sigma_g^b)) = \langle \lambda \rangle\) is Abelian.

    \item[\bfseries\color{highlight}(2)]
      In this case, \(W = \ker (L_{\alpha_g} - \lambda)^2\) is a
      \(2\)-dimensional \(\Mod(\Sigma)\)-invariant subspace.
  \end{enumerate}

  For cases (3) and (4), we consider two situations: \(E_{\alpha_g = \lambda}
  \ne E_{\beta_g = \lambda}\) or \(E_{\alpha_g = \lambda} = E_{\beta_g =
  \lambda}\). If \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\), then \(W
  = E_{\alpha_g = \lambda} \cap E_{\beta_g = \lambda}\) is a \((n -
  2)\)-dimensional \(\Mod(\Sigma)\)-invariant subspace: since \(\beta_g\) lies
  outside of \(\Sigma\) and \(L_{\alpha_g}, L_{\beta_g}\) are conjugate, both
  \(E_{\alpha_g = \lambda}\) and \(E_{\beta_g = \lambda}\) are
  \(\Mod(\Sigma)\)-invariant \((n - 1)\)-dimensional subspaces.

  Finally, we consider the case where \(E_{\alpha_g = \lambda} = E_{\beta_g
  = \lambda}\). In this situation, as in the proof of
  Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}, it follows from
  Observation~\ref{ex:change-of-coordinates-crossing} that there are \(f_i,
  g_i, h_i \in \Mod(\Sigma_g^b)\) with
  \begin{align*}
    f_i \tau_{\alpha_g}    f_i^{-1} & = \tau_{\alpha_i}
    &
    g_i \tau_{\alpha_g}    g_i^{-1} & = \tau_{\beta_i}
    &
    h_i \tau_{\alpha_g}    h_i^{-1} & = \tau_{\alpha_g}
    \\
    f_i \tau_{\beta_g} f_i^{-1} & = \tau_{\beta_i}
    &
    g_i \tau_{\beta_g} g_i^{-1} & = \tau_{\gamma_i}
    &
    h_i \tau_{\beta_g} h_i^{-1} & = \tau_{\eta_i}
  \end{align*}
  and thus
  \(E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda} = E_{\beta_1 =
  \lambda} = \cdots = E_{\beta_g = \lambda} = E_{\gamma_1 = \lambda} = \cdots =
  E_{\gamma_{g - 1} = \lambda} = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1}
  = \lambda}\).

  In particular, we can find a basis for \(\mathbb{C}^n\) with respect to
  which the matrix of any Lickorish generator has the form
  \[
    \begin{pmatrix}
      \lambda & 0       & \cdots & 0       & *      \\
      0       & \lambda & \ldots & 0       & *      \\
      \vdots  & \vdots  & \ddots & \vdots  & \vdots \\
      0       & 0       & \cdots & \lambda & *      \\
      0       & 0       & \cdots & 0       & *
    \end{pmatrix}.
  \]
  Since the group of upper triangular matrices is solvable and
  \(\Mod(\Sigma_g^b)\) is perfect, it follows that \(\rho(\Mod(\Sigma_g^b))\) is
  trivial. This concludes the proof \(\rho(\Mod(\Sigma_g^b))\) is Abelian.

  To see that \(\rho(\Mod(\Sigma_g^b)) = 1\) for \(g \ge 3\) we note that,
  since \(\rho\) has Abelian image and thus factors though the Abelianization
  map \(\Mod(\Sigma_g^b) \to \Mod(\Sigma_g^b)^\ab =
  \mfrac{\Mod(\Sigma_g^b)}{[\Mod(\Sigma_g^b), \Mod(\Sigma_g^b)]}\). Now recall
  from Proposition~\ref{thm:trivial-abelianization} that \(\Mod(\Sigma_g^b)^\ab
  = 0\) for \(g \ge 3\). We are done.
\end{proof}

Having established the triviality of the low-dimensional representations \(\rho
: \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to
understand the \(2g\)-dimensional representations of \(\Mod(\Sigma_g^b)\). We
certainly know a nontrivial example of such, namely the symplectic
representation \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\)
from Example~\ref{ex:symplectic-rep}. Surprisingly, this turns out to be
\emph{essentially} the only example of a nontrivial \(2g\)-dimensional
representation in the compact case. More precisely,

\begin{theorem}[Korkmaz]\label{thm:reps-of-dim-2g-are-symplectic}
  Let \(g \ge 3\) and \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\).
  Then \(\rho\) is either trivial or conjugate to the symplectic
  representation\footnote{Here the map $\Mod(\Sigma_g^b) \to
  \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the
  inclusion morphism $\Mod(\Sigma_g^b) \to \Mod(\Sigma_g)$ with the usual
  symplectic representation $\psi : \Mod(\Sigma_g) \to
  \operatorname{Sp}_{2g}(\mathbb{Z})$.} \(\Mod(\Sigma_g^b) \to
  \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^b)\).
\end{theorem}

Unfortunately, the limited scope of these master's thesis does not allow us to
dive into the proof of Theorem~\ref{thm:reps-of-dim-2g-are-symplectic}. The
heart of this proof lies in a result about representations of the product
\(B_3^n = B_3 \times \cdots \times B_3\), which Korkmaz refers to as \emph{the
main lemma}.

\begin{lemma}[Korkmaz' main lemma]\label{thm:main-lemma}
  Given \(i = 1, \ldots, n\), denote by 
  \begin{align*}
    a_i & = (1, \ldots, 1, \sigma_1, 1, \ldots, 1) &
    b_i & = (1, \ldots, 1, \sigma_2, 1, \ldots, 1)
  \end{align*}
  the \(n\)-tuples in \(B_3^n\) whose \(i\)-th coordinates are \(\sigma_1\) and
  \(\sigma_2\), respectively, and with all other coordinates equal to \(1\).
  Let \(m \ge 2n\) and \(\rho : B_3^n \to \GL_m(\mathbb{C})\) be a
  representation satisfying:
  \begin{enumerate}
    \item The only eigenvalue of \(\rho(a_i)\) is \(1\) and its eigenspace is
      \((m - 1)\)-dimensional.
    \item The eigenspaces of \(\rho(a_i)\) and \(\rho(b_i)\) associated to the
      eigenvalue \(1\) do not coincide.
  \end{enumerate}
  Then \(\rho\) is conjugate to the representation
  \begin{align*}
    B_3^n & \to \GL_m(\mathbb{C}) \\
    a_i
    & \mapsto
    \left(
    \begin{array}{c|c|c}
      1_{2(i-1)} & 0                                            & 0 \\ \hline
               0 & \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} & 0 \\ \hline
               0 & 0                                            & 1_{m-2i}
    \end{array}
    \right) \\
    b_i
    & \mapsto
    \left(
    \begin{array}{c|c|c}
      1_{2(i-1)} & 0                                             & 0 \\ \hline
               0 & \begin{array}{cc} 1 & 0 \\ -1 & 1 \end{array} & 0 \\ \hline
                 0 & 0                                             & 1_{m-2i}
    \end{array}
    \right),
  \end{align*}
  where \(1_k\) denotes the \(k \times k\) identity matrix.
\end{lemma}

This is proved in \cite[Lemma 7.6]{korkmaz} using the braid relations. Notice
that for \(n = g\) and \(m = 2g\) the matrices in Lemma~\ref{thm:main-lemma}
coincide with the action of the Lickorish generators \(\tau_{\alpha_1}, \ldots,
\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(\Sigma_g^b)\) on
\(H_1(\Sigma_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard
basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for
\(H_1(\Sigma_g, \mathbb{C})\).
\begin{align*}
  (\tau_{\alpha_i})_* & =
  \left(
  \begin{array}{c|c|c}
    1 & 0                                            & 0 \\ \hline
    0 & \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} & 0 \\ \hline
    0 & 0                                            & 1
  \end{array}
  \right) &
  (\tau_{\beta_i})_* & =
  \left(
  \begin{array}{c|c|c}
    1 & 0                                             & 0 \\ \hline
    0 & \begin{array}{cc} 1 & 0 \\ -1 & 1 \end{array} & 0 \\ \hline
    0 & 0                                             & 1
  \end{array}
  \right)
\end{align*}

Hence by embedding \(B_3^g\) in \(\Mod(\Sigma_g^b)\) via
\begin{align*}
  B_3^g & \to \Mod(\Sigma_g^b)         \\
  a_i   & \mapsto \tau_{\alpha_i}    \\
  b_i   & \mapsto \tau_{\beta_i}
\end{align*}
we can see that any \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\) in a
certain class of representation satisfying some technical conditions must be
conjugate to the symplectic representation \(\Mod(\Sigma_g^b) \to
\operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\).

Korkmaz then goes on to show that such technical conditions are met for any
nontrivial \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\). Furthermore,
Korkmaz also argues that we can find a basis for \(\mathbb{C}^{2g}\) with
respect to which the matrices of \(\rho(\tau_{\gamma_1}), \ldots,
\rho(\tau_{\gamma_{g - 1}}), \rho(\tau_{\eta_1}), \ldots,
\rho(\tau_{\eta_{b-1}})\) also agree with the action of \(\Mod(\Sigma_g^b)\) on
\(H_1(\Sigma_g, \mathbb{C})\), concluding the classification of
\(2g\)-dimensional representations.

Recently, Kasahara \cite{kasahara} also classified the \((2g+1)\)-dimensional
representations of \(\Mod(\Sigma_g^b)\) for \(g \ge 7\) in terms of certain
twisted \(1\)-cohomology groups. On the other hand, the representations of
dimension \(n > 2g + 1\) are still poorly understood, and fundamental questions
remain unsweared. In the short and mid-terms, the works of Korkmaz and Kasahara
lead to many follow-up questions. For example,
\begin{enumerate}
  \item In the \(g \ge 3\) case, Korkmaz \cite[Theorem~3]{korkmaz} established
    the lower bound of \(3g - 3\) for the dimension of an injective linear
    representation of \(\Mod(\Sigma_g)\) -- if one such representation exists.
    Can we improve this lower bound?

  \item What is the minimal dimension for a representation of
    \(\Mod(\Sigma_g)\) which does not annihilate the entire kernel of the
    symplectic representation \(\psi : \Mod(\Sigma_g) \to
    \operatorname{Sp}_{2g}(\mathbb{Z})\)? In particular, do the \((2g +
    1)\)-dimensional representations classified by Kasahara \cite{kasahara}
    annihilate all of \(\ker \psi\)?
\end{enumerate}

These are some of the questions which I plan to work on during my upcoming PhD.
