diff --git a/sections/representations.tex b/sections/representations.tex
@@ -1,50 +1,59 @@
\chapter{Low-Dimensional Representations}
-\begin{theorem}[Korkmaz \cite{korkmaz}]\label{thm:low-dim-reps-are-trivial}
+Having built a solid understanding of the combinatorics of Dehn twists, we are
+now ready to attack the problem of classifying the representations of
+\(\Mod(S_g)\). Indeed, in light of the Wajnryb presentation, a representation
+\(\rho : \Mod(S_g) \to \GL_n(\mathbb{C})\) is nothing other than a choice of
+\(2g + 1\) matrices \(\rho(\tau_{\alpha_1}), \ldots, \rho(\tau_{\alpha_{2g}})
+\in \GL_n(\mathbb{C})\) satisfying the relations \strong{(i)} to \strong{(v)}
+from Theorem~\ref{thm:wajnryb-presentation}.
+
+Historically, these relations have been exploited by Funar \cite{funar},
+Franks-Handel \cite{franks-handel} and others to establish the triviality of
+low-dimensional representions, culminating Korkmaz' recent classification of
+representations of dimension \(n \le 2 g\) for \(g \ge 3\) \cite{korkmaz}. The
+goal of this chapter is providing a concise account of Korkmaz' results,
+starting by\dots
+
+\begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial}
Let \(S_g^b\) be the surface of genus \(g \ge 1\) and \(b\) boundary
- components and \(\rho : \Mod(S_g^b) \to \GL(V)\) be an \(m\)-dimensional
- linear representation for some \(m < 2 g\). Then the image of \(\rho\) is
- Abelian. In particular, if \(g \ge 3\) then \(\rho\) is trivial.
+ components and \(\rho : \Mod(S_g^b) \to \GL_n(\mathbb{C})\) 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}
-% TODO: Explain the setup of the proof: induction in m and case analysis on the
-% Jordan form
+Like so many 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 \(g
+= 2\).
-% TODO: Explain this is the base case
\begin{proposition}\label{thm:low-dim-reps-are-trivial-base-case}
- Let \(\rho : \Mod(S_2^b) \to \GL(V)\) be an \(m\)-dimensional representation,
- \(m \le 3\). Then the image of \(\rho\) is a quotient of \(\mathbb{Z}/10\).
+ Given \(\rho : \Mod(S_2^b) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the
+ image of \(\rho\) is Abelian.
\end{proposition}
-% I don't think it's worth including the whole proof in here: the case analysis
-% is too boring and takes too much space
\begin{proof}[Sketch of proof]
- % TODO: You haven't commented on the Abelianization in the case with boundary
- % It is easy to see that Mod(S_2^b)^ab is a quotient of ℤ/10: the map
- % Mod(S_2^b)^ab → Mod(S_2)^ab induced by the inclusion morphism is surjective
- Since \(\Mod(S_2^b)^\ab \cong \mathbb{Z}/10\), it suffices to show
- \(\rho(\Mod(S_2^b))\) is Abelian, so that \(\rho\) factors through the
- Abelianization map \(\Mod(S_2^b) \to \Mod(S_2^b)^\ab\). Equivalenty, it
- suffices to show that \(\rho(\Mod(S_2^b)') = 1\). Given \(\alpha \subset
- S_2^b\), denote \(L_\alpha = \rho(\tau_\alpha)\). Let \(\alpha_1,
- \alpha_2, \mu_1, \mu_2, \gamma, \eta_1, \ldots, \eta_{b - 1} \subset
- S_2^b\) be the curves of the Lickorish generators from
- Theorem~\ref{thm:lickorish-gens}.
+ Given \(\alpha \subset S_2^b\), 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 \(\alpha_1, \alpha_2, \mu_1, \mu_2,
+ \gamma, \eta_1, \ldots, \eta_{b - 1} \subset S_2^b\) be the curves of the
+ Lickorish generators from Theorem~\ref{thm:lickorish-gens}.
\begin{center}
\includegraphics[width=.25\linewidth]{images/lickorish-gens-gen-2.eps}
\end{center}
- If \(m = 1\) then \(\GL(V) = \mathbb{C}^\times\) is Abelian and hence so is
- \(\rho(\Mod(S_2^b))\). Now if \(m = 2\) or \(3\), by
+ If \(n = 1\) then \(\rho(\Mod(S_2^b)) \subset \GL_1(\mathbb{C}) =
+ \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by
Propositon~\ref{thm:commutator-normal-gen} it suffices to show \(L_{\alpha_1}
= L_{\mu_1}\), so that \(\tau_{\alpha_1} \tau_{\mu_1}^{-1} \in \ker \rho\)
- and thus \(\Mod(S_2^b)' \subset \ker \rho\). Given the braid relation
+ and thus \(\Mod(S_2^b)' \subset \ker \rho\) -- i.e. \(\rho(\Mod(S_2^b))\) is
+ Abelian. Given the braid relation
\begin{equation}\label{eq:braid-rel-induction-basis}
L_{\alpha_1} L_{\mu_1} L_{\alpha_1} = L_{\mu_1} L_{\alpha_1} L_{\mu_1},
\end{equation}
this amounts to showing \(L_{\alpha_1}\) and \(L_{\mu_1}\) commute.
- To that end, we exhausively analyse all of the possible Jordan decompositions
+ To that end, we exhausively analyse all of the possible Jordan forms
\begin{align*}
\begin{pmatrix}
\lambda & 0 \\
@@ -106,25 +115,25 @@
\end{pmatrix}
& \quad{\normalfont(9)}
\end{align*}
- of \(L_{\mu_2}\) in some basis \(\mathcal{B}\) -- where \(\lambda, \mu,
- \nu \in \mathbb{C}^\times\) are all distinct.
+ of \(L_{\mu_2}\) -- where \(\lambda, \mu, \nu \in \mathbb{C}^\times\) are all
+ distinct. By changing basis we may assume without loss of generality that the
+ matrix \(L_{\mu_2}\) is exactly its Jordan form, so that \(E_{\mu_2 =
+ \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\).
For cases (1) to (7) we use the change of coordinates principle and the braid
- relation (\ref{eq:braid-rel-induction-basis}) to show that the matrices of
- \(L_{\alpha_1}\) and \(L_{\mu_1}\) in the basis \(\mathcal{B}\) lie in
- some Abelian subgroup of \(\GL_m(\mathbb{C})\), \(m = 2\) or \(3\) -- hence
- they commute. See \cite[Proposition~5.1]{korkmaz} for further details.
- For cases (8) and (9) we consider the curve \(\alpha_2\). In these cases,
- the eigenspace \(V_{L_{\mu_2} = \lambda}\) is \(2\)-dimensional. Since
- \(L_{\mu_2}\) and \(L_{\alpha_2}\) are conjugate, so is \(V_{L_{\alpha_2}
- = \lambda}\) -- indeed, conjugate operators have the same Jordan form. Now
- either \(V_{L_{\mu_2} = \lambda} = V_{L_{\alpha_2} = \lambda}\) or
- \(V_{L_{\mu_2} = \lambda} \ne V_{L_{\alpha_2} = \lambda}\). We begin by
- the first case.
-
- We claim that if \(V_{L_{\mu_2} = \lambda} = V_{L_{\alpha_2} = \lambda}\)
- then \(V_{L_{\alpha_0} = \lambda}\) is \(\Mod(S_2^b)\)-invariant. Indeed, by
- change of coordinates we can always find \(f, g, h_i \in \Mod(S_2^b)\) with
+ relation (\ref{eq:braid-rel-induction-basis}) to show that \(L_{\alpha_1}\)
+ and \(L_{\mu_1}\) lie in some Abelian subgroup of \(\GL_n(\mathbb{C})\) --
+ hence they commute. See \cite[Proposition~5.1]{korkmaz} for further details.
+ For cases (8) and (9) we consider the curve \(\alpha_2\). In these cases, the
+ eigenspace \(E_{\mu_2 = \lambda}\) is \(2\)-dimensional. Since \(L_{\mu_2}\)
+ and \(L_{\alpha_2}\) are conjugate, \(E_{\alpha_2 = \lambda}\) is also
+ \(2\)-dimensional -- indeed, conjugate operators have the same Jordan form.
+ Now either \(E_{\mu_2 = \lambda} = E_{\alpha_2 = \lambda}\) or \(E_{\mu_2 =
+ \lambda} \ne E_{\alpha_2 = \lambda}\). We begin by the first case.
+
+ We claim that if \(E_{\mu_2 = \lambda} = E_{\alpha_2 = \lambda}\)
+ then \(E_{\mu_2 = \lambda}\) is \(\Mod(S_2^b)\)-invariant. Indeed, by change
+ of coordinates we can always find \(f, g, h_i \in \Mod(S_2^b)\) with
\begin{align*}
f \cdot [\mu_2] & = [\mu_1]
&
@@ -152,136 +161,141 @@
\end{align*}
and thus
\begin{align*}
- V_{L_{\mu_1} = \lambda}
- = \rho(f) V_{L_{\mu_2} = \lambda}
- & = \rho(f) V_{L_{\alpha_2} = \lambda}
- = V_{L_{\alpha_1} = \lambda}
+ E_{\mu_1 = \lambda}
+ = \rho(f) E_{\mu_2 = \lambda}
+ & = \rho(f) E_{\alpha_2 = \lambda}
+ = E_{\alpha_1 = \lambda}
\\
- V_{L_{\alpha_1} = \lambda}
- = \rho(g) V_{L_{\mu_2} = \lambda}
- & = \rho(g) V_{L_{\alpha_2} = \lambda}
- = V_{L_\gamma = \lambda}
+ E_{\alpha_1 = \lambda}
+ = \rho(g) E_{\mu_2 = \lambda}
+ & = \rho(g) E_{\alpha_2 = \lambda}
+ = E_{\gamma = \lambda}
\\
- V_{L_{\eta_i} = \lambda}
- = \rho(h_i) V_{L_{\mu_2} = \lambda}
- & = \rho(h_i) V_{L_{\alpha_2} = \lambda}
- = V_{L_{\alpha_2} = \lambda}.
+ E_{\eta_i = \lambda}
+ = \rho(h_i) E_{\mu_2 = \lambda}
+ & = \rho(h_i) E_{\alpha_2 = \lambda}
+ = E_{\alpha_2 = \lambda}.
\end{align*}
- In other words, \(V_{L_{\alpha_1} = \lambda} = V_{L_{\alpha_2} = \lambda} =
- V_{L_{\mu_1} = \lambda} = V_{L_{\mu_2} = \lambda} = V_{L_\gamma = \lambda} =
- V_{L_{\eta_1} = \lambda} = \cdots = V_{L_{\eta_{b - 1}} = \lambda}\) is
+ In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} =
+ E_{\mu_1 = \lambda} = E_{\mu_2 = \lambda} = E_{\gamma = \lambda} =
+ E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b - 1} = \lambda}\) is
invariant under the action of all Lickorish generators.
Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : \Mod(S_2^b) \to
- \GL(V_{L_{\mu_2} = \lambda})\). By case (2), \(\bar \rho(f) = 1\) for all
- \(f \in \Mod(S_2^b)'\), for \(\bar \rho(\Mod(S_2^b))\) is Abelian. In other
- words, the matrix of \(\rho(f)\) in the basis \(\mathcal{B}\) has the form
+ \GL(E_{\mu_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\mu_2 =
+ \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By case (2), \(\bar
+ \rho(f) = 1\) for all \(f \in \Mod(S_2^b)'\), given that \(\bar
+ \rho(\Mod(S_2^b))\) is Abelian. Thus
\[
+ \rho(\Mod(S_2^b)') \subset
\begin{pmatrix}
1 & 0 & * \\
0 & 1 & * \\
0 & 0 & *
\end{pmatrix}
\]
- and, in particular, it lies inside the group of upper triangular matrices --
- a solvalbe subgroup of \(\GL_3(\mathbb{C})\). Now by
- Proposition~\ref{thm:commutator-is-perfect} we get \(\rho(\Mod(S_2^b)') =
- 1\): any homomorphism from a perfect group to a solvable group is trivial.
+ lies inside the group of upper triangular matrices, a solvalbe subgroup of
+ \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we
+ get \(\rho(\Mod(S_2^b)') = 1\): any homomorphism from a perfect group to a
+ solvable group is trivial.
- Finally, if \(V_{L_{\mu_2} = \lambda} \ne V_{L_{\alpha_2} = \lambda}\) and
+ Finally, if \(E_{\mu_2 = \lambda} \ne E_{\alpha_2 = \lambda}\) and
the Jordan form of \(L_{\mu_2}\) is given by (8) then
\[
0
- \subsetneq V_{L_{\mu_2} = \lambda} \cap V_{L_{\alpha_2} = \lambda}
- \subsetneq V_{L_{\mu_2} = \lambda}
+ \subsetneq E_{\mu_2 = \lambda} \cap E_{\alpha_2 = \lambda}
+ \subsetneq E_{\mu_2 = \lambda}
\subsetneq V
\]
is a flag of subspaces invariant under \(L_{\mu_1}\) and \(L_{\alpha_1}\),
for \(\mu_2\) is disjoint from \(\mu_1 \cup \alpha_1\) and thus
\([\tau_{\mu_2}, \tau_{\mu_1}] = [\tau_{\mu_2}, \tau_{\alpha_1}] = 1\). In
- this case we can find a basis \(\mathcal{B}'\) for \(V\) in wich the matrices
- of \(L_{\mu_1}\) and \(L_{\alpha_1}\) are both upper triangular with
- \(\lambda\) along the diagonal: take \(\mathcal{B}' = \{v_1, v_2, v_3\}\)
- with \(v_1 \in V_{L_{\mu_2} = \lambda} \cap V_{L_{\alpha_2} = \lambda}\),
- \(v_2 \in V_{L_{\mu_2}}\) and adjust \(v_3\) to get the desired diagonal
- entry. Any such pair of matrices satisfying the braid relation
+ this case we can find a basis for \(\mathbb{C}^3\) with respect to wich the
+ matrices of \(L_{\mu_1}\) and \(L_{\alpha_1}\) are both upper triangular with
+ \(\lambda\) along the diagonal: take \(v_1, v_2, v_3 \in \mathbb{C}^3\) with
+ \(v_1 \in E_{\mu_2 = \lambda} \cap E_{\alpha_2 = \lambda}\), \(v_2 \in
+ V_{L_{\mu_2}}\) and adjust \(v_3\) to get the desired diagonal entry. Any
+ such pair of matrices satisfying the braid relation
(\ref{eq:braid-rel-induction-basis}) commute.
- Similarly, if \(L_{\mu_2}\) has Jordan form (9) and \(V_{L_{\mu_2} = \lambda}
- \ne V_{L_{\alpha_2} = \lambda}\) we use (\ref{eq:braid-rel-induction-basis})
- to conclude \([L_{\mu_1}, L_{\alpha_1}] = 1 \in \GL(V)\) -- again, see
- \cite[Proposition~5.1]{korkmaz} for further details. We are done.
+ Similarly, if \(L_{\mu_2}\) has Jordan form (9) and \(E_{\mu_2 = \lambda}
+ \ne E_{\alpha_2 = \lambda}\) we use (\ref{eq:braid-rel-induction-basis})
+ to conclude \(L_{\mu_1}\) and \(L_{\alpha_1}\) commute -- again, see
+ \cite[Proposition~5.1]{korkmaz}. We are done.
\end{proof}
-\begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}]
- Let \(g \ge 3\), \(b \ge 0\) and \(\rho : \Mod(S_g^b) \to \GL(V)\) be an
- \(m\)-dimensional representation with \(m < 2g\).
+We are now ready to establish the triviality of low-dimensional
+representations.
- As promised, we proceed by induction on \(g\). If \(g = 1\) then \(m = 1\)
- and thus \(\rho(\Mod(S_g^b)) \subset \GL(V) = \mathbb{C}^\times\) is Abelian.
- The base case \(g = 2\) was also established in
- Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}. Now suppose \(g \ge
- 3\) and every \(n\)-dimensional representation of \(S_{g - 1}^c\) has Abelian
- image for \(n < 2(g - 1)\) and \(c \ge 0\). Let us show \(\rho\) has Abelian
- image.
+\begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}]
+ Let \(g \ge 1\), \(b \ge 0\) and fix \(\rho : \Mod(S_g^b) \to
+ \GL_n(\mathbb{C})\) with \(n < 2g\). 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 \(S_{g -
+ 1}^{b'}\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has
+ Abelian image.
Let \(\alpha_1, \ldots, \alpha_g, \mu_1, \ldots, \mu_g, \gamma_1, \ldots,
\gamma_{g - 1}, \beta_1, \ldots, \beta_{b - 1} \subset S_g^b\) be the curves
from the Lickorish generators of \(\Mod(S_g^b)\), as in
- Theorem~\ref{thm:lickorish-gens}. Once again, denote \(L_\alpha =
- \rho(\tau_\alpha)\) for any closed \(\alpha \subset S_g^b\). Let \(R \cong
- S_{g - 1}^1\) be the closed subsurface highlighted in the following picture.
+ Theorem~\ref{thm: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 \(R \cong S_{g -
+ 1}^1\) be the closed subsurface highlighted in the following picture.
\begin{center}
\includegraphics[width=.5\linewidth]{images/lickorish-gens-korkmaz-proof.eps}
\end{center}
% TODO: Add more comments on the injectivity of this map?
- We claim that it suffices to find a \(\Mod(R)\)-invariant\footnote{Here we
- view $\Mod(R)$ as a subgroup of $\Mod(S_g^b)$ via the inclusion homomorphism
- $\Mod(R) \to \Mod(S_g^b)$ from Example~\ref{ex:inclusion-morphism}, which can
- be shown to be injective in this particular case.} subspace \(W \subset V\)
- of dimension \(n\) with \(2 \le n \le m - 2\). Indeed, in this case \(n < 2(g
- - 1)\) and \(\dim \mfrac{V}{W} = m - n < 2(g - 1)\). Thus both
- representations
+ We claim that it suffices to find a \(m\)-dimensional
+ \(\Mod(R)\)-invariant\footnote{Here we view $\Mod(R)$ as a subgroup of
+ $\Mod(S_g^b)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^b)$ 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(R) & \to \GL(W) & \rho_2 : \Mod(R) & \to \GL(\mfrac{V}{W})
+ \rho_1 : \Mod(R) & \to \GL(W) \cong \GL_m(\mathbb{C})
+ &
+ \rho_2 : \Mod(R) & \to \GL(\mfrac{\mathbb{C}^n}{W})
+ \cong \GL_{n - m}(\mathbb{C})
\end{align*}
fall into the induction hypotesis -- i.e. \(\rho_i(\Mod(R))\) is
Abelian. In particular, \(\rho_i(\Mod(R)') = 1\) and we can find some
- basis for \(V\) under which
+ basis for \(\mathbb{C}^n\) with respect to which
\[
- % TODO: Make this prettier: somehow format the block sizes in the equation
\rho(f) =
\left(
\begin{array}{c|c}
- 1 & * \\ \hline
- 0 & 1
+ 1_m & * \\ \hline
+ 0 & 1_{n - m}
\end{array}
\right)
\]
- for any \(f \in \Mod(R)'\) -- where the first and second diaogonal blocks are
- \(n \times n\) and \((m - n) \times (m - n)\). Since the group of upper
- triangular matrices is solvable, it follows from
- Proposition~\ref{thm:commutator-is-perfect} that \(\rho\) annihilates all
- \(\Mod(R)'\) and, in particular, \(\tau_{\alpha_1} \tau_{\mu_1}^{-1} \in \ker
- \rho\). But recall from Proposition~\ref{thm:commutator-normal-gen} that
- \(\Mod(S_g^b)'\) is normally generated by \(\tau_{\alpha_1}
- \tau_{\mu_1}^{-1}\), from which we conclude \(\rho(\Mod(S_g^b)') = 1\), as
- desired.
-
- As before, we exhaustively analyse all possible Jordan forms for
+ for all \(f \in \Mod(R)'\) -- 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 \(\Mod(R)'\) and, in particular, \(\tau_{\alpha_1}
+ \tau_{\mu_1}^{-1} \in \ker \rho\). But recall from
+ Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(S_g^b)'\) is normally
+ generated by \(\tau_{\alpha_1} \tau_{\mu_1}^{-1}\), from which we conclude
+ \(\rho(\Mod(S_g^b)') = 1\), as desired.
+
+ As before, we exhaustively analyse all possible Jordan forms of
\(L_{\mu_g}\). First, consider the case where we can find eigenvalues
\(\lambda_1, \ldots, \lambda_k\) of \(L_{\mu_g}\) such that the sum \(W =
- \bigoplus_i V_{L_{\mu_g} = \lambda_i}\) of the corresponding eigenspaces has
- dimension \(n\) with \(2 \le n \le m - 2\). In this case, it suffices to
- observe that since \(\mu_g\) lies outside of \(R\), each \(V_{L_{\mu_g} =
+ \bigoplus_i E_{\mu_g = \lambda_i}\) of the corresponding eigenspaces has
+ dimension \(m\) with \(2 \le m \le n - 2\). In this case, it suffices to
+ observe that since \(\mu_g\) lies outside of \(R\), each \(E_{\mu_g =
\lambda_i}\) is \(\Mod(R)\)-invariant: the Lickorish generators
\(\tau_{\alpha_1}, \ldots, \tau_{\alpha_{g - 1}}, \tau_{\mu_1}, \ldots,
- \tau_{\mu_{g - 1}}, \tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - 2}}\) of \(R
- \cong S_{g - 1}^1\) all commute with \(\tau_{\mu_g}\) and thus preserve the
- eigenspaces of it's action on \(V\).
+ \tau_{\mu_{g - 1}}\), \(\tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - 2}}\) of
+ \(R \cong S_{g - 1}^1\) all commute with \(\tau_{\mu_g}\) and thus preserve
+ the eigenspaces of its action on \(\mathbb{C}^n\).
- If no sum of the form \(\bigoplus_i V_{L_{\mu_g} = \lambda_i}\) has
+ If no sum of the form \(\bigoplus_i E_{\mu_g = \lambda_i}\) has
dimension lying between \(2\) and \(m - 2\) there must be at most \(2\)
distinct eigenvalues and all eigenspaces must be either \(1\)-dimensional or
\((m - 1)\)-dimensional. Hence then the Jordan form of \(L_{\mu_g}\) has to
@@ -323,29 +337,27 @@
\end{pmatrix}
& \quad{\normalfont(4)}
\end{align*}
- for \(\lambda \ne \mu\). We now analyze each one of these sporadic cases
+ for \(\lambda \ne \mu\). We analyze each one of these sporadic cases
individually.
For case (1), we use the change of coordinates principle: each
\(L_{\alpha_i}, L_{\mu_i}, L_{\gamma_i}, L_{\eta_i}\) is conjugate to
\(L_{\mu_g} = \lambda\), so all Lickorish generators of \(\Mod(S_g^b)\) act
- on \(V\) as scalar multiplication by \(\lambda\) as well. Hence
- \(\rho(\Mod(S_g^b))\) is cyclic and thus Abelian. In case (2) \(W = \ker
- (L_{\alpha_{2 - g}} - \lambda)^2\) is a \(2\)-dimensional
- \(\Mod(R)\)-invariant subspace.
-
- For cases (3) and (4) we consider two situations: \(V_{L_{\mu_g} =
- \lambda} \ne V_{L_{\alpha_g} = \lambda}\) or \(V_{L_{\mu_g} =
- \lambda} = V_{L_{\alpha_g} = \lambda}\). In the first case, \(W =
- V_{L_{\mu_g} = \lambda} \cap V_{L_{\alpha_g} = \lambda}\) is a
- \((m - 2)\)-dimensional \(\Mod(R)\)-invariant subspace: since \(L_{\alpha_{2
- - g}}\) and \(L_{\alpha_g}\) are conjugate and \(\alpha_g\) lies
- outside of \(R\), both \(V_{L_{\mu_g} = \lambda}\) and
- \(V_{L_{\alpha_g} = \lambda}\) are \(\Mod(R)\)-invariant \((m -
- 1)\)-dimensional subspaces.
-
- Finally, we consider the case where \(V_{L_{\mu_g} = \lambda} =
- V_{L_{\alpha_g} = \lambda}\). In this situation, as in the proof of
+ on \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence
+ \(\rho(\Mod(S_g^b))\) is cyclic and thus Abelian. In case (2), \(W = \ker
+ (L_{\mu_g} - \lambda)^2\) is a \(2\)-dimensional \(\Mod(R)\)-invariant
+ subspace.
+
+ For cases (3) and (4) we consider two situations: \(E_{\mu_g = \lambda} \ne
+ E_{\alpha_g = \lambda}\) or \(E_{\mu_g = \lambda} = E_{\alpha_g = \lambda}\).
+ In the first case, \(W = E_{\mu_g = \lambda} \cap E_{\alpha_g = \lambda}\) is
+ a \((m - 2)\)-dimensional \(\Mod(R)\)-invariant subspace: since \(L_{\mu_g}\)
+ and \(L_{\alpha_g}\) are conjugate and \(\alpha_g\) lies outside of \(R\),
+ both \(E_{\mu_g = \lambda}\) and \(E_{\alpha_g = \lambda}\) are
+ \(\Mod(R)\)-invariant \((m - 1)\)-dimensional subspaces.
+
+ Finally, we consider the case where \(E_{\mu_g = \lambda} =
+ E_{\alpha_g = \lambda}\). In this situation, as in the proof of
Proposition~\ref{thm:low-dim-reps-are-trivial-base-case} it follows from the
change of coordinates principle that there are \(f_i, g_i, h_i \in
\Mod(S_g^b)\) with
@@ -364,14 +376,14 @@
\end{align*}
and thus
\[
- V_{L_{\alpha_1} = \lambda} = \cdots = V_{L_{\alpha_g} = \lambda}
- = V_{L_{\mu_1} = \lambda} = \cdots = V_{L_{\mu_g} = \lambda}
- = V_{L_{\gamma_1} = \lambda} = \cdots = V_{L_{\gamma_{g - 1}} = \lambda}
- = V_{L_{\eta_1} = \lambda} = \cdots = V_{L_{\eta_{b - 1}} = \lambda}.
+ E_{\mu_1 = \lambda} = \cdots = E_{\mu_g = \lambda}
+ = E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda}
+ = E_{\gamma_1 = \lambda} = \cdots = E_{\gamma_{g - 1} = \lambda}
+ = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b - 1} = \lambda}.
\]
- In particular, we can find a basis for \(V\) under which the matrix of all
- Lickorish generators has the form
+ In particular, we can find a basis for \(\mathbb{C}^n\) with respect to which
+ the matrix of any Lickorish generators has the form
\[
\begin{pmatrix}
\lambda & 0 & \cdots & 0 & * \\
@@ -382,15 +394,128 @@
\end{pmatrix}.
\]
Since the group of upper triangular matrices is solvable and \(\Mod(S_g^b)\)
- is perfect, it follows that \(\rho(\Mod(S_g^b))\) is trivial. We are done.
+ is perfect, it follows that \(\rho(\Mod(S_g^b))\) is trivial. This concludes
+ the proof \(\rho(\Mod(S_g^b))\) is Abelian.
+
+ To see that \(\rho(\Mod(S_g^b)) = 1\) for \(g \ge 3\) we note that, since
+ \(\rho(\Mod(S_g^b))\) is Abelian, \(\rho\) factors though the Abelinization
+ map \(\Mod(S_g^b) \to \Mod(S_g^b)^\ab = \mfrac{\Mod(S_g^b)}{[\Mod(S_g^b),
+ \Mod(S_g^b)]}\) Now recall from Proposition~\ref{thm:trivial-abelianization}
+ that \(\Mod(S_g^b)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\)
+ factors though the homomorphism \(1 \to \GL_n(\mathbb{C})\). We are done.
\end{proof}
-\begin{theorem}[Korkmaz \cite{korkmaz}]
- Let \(g \ge 3\) and \(\rho : \Mod(S_g^b) \to \GL(V)\) be a \(2g\)-dimensional
- linear representation. Then either \(\rho\) is either trivial or conjugate to
- the symplectic representation\footnote{Here the map \(\Mod(S_g^b) \to
- \operatorname{Sp}_{2g}(\mathbb{Z})\) is given by the composition of the
- inclusion morphism \(\Mod(S_g^b) \to \Mod(S_g)\) with the usual symplect
- representation \(\psi : \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\).}
+Having established the triviality of the low-dimensional representations \(\rho
+: \Mod(S_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to understand
+the \(2g\)-dimensional reprensentations of \(\Mod(S_g^b)\). We certainly know a
+nontrivial example of such, namely the symplectic representation \(\psi :
+\Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from
+Example~\ref{ex:symplectic-rep}. Surprinsgly, 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 \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\). Then
+ \(\rho\) is either trivial or conjugate to the symplectic
+ representation\footnote{Here the map $\Mod(S_g^b) \to
+ \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the
+ inclusion morphism $\Mod(S_g^b) \to \Mod(S_g)$ with the usual symplect
+ representation $\psi : \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.}
\(\Mod(S_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(S_g^b)\).
\end{theorem}
+
+Unfortunately, the limited scope of these master 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 somewhat technical 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}. Namely\dots
+
+\begin{lemma}[Korkmaz' Main Lemma]\label{thm:main-lemma}
+ Given \(i = 1, \ldots, n\), denote by \newline \(a_i = (1, \ldots, 1,
+ \sigma_1, 1, \ldots 1)\) and \(b_i = (1, \ldots, 1, \sigma_2, 1, \ldots, 1)\)
+ 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 it's eigenspace is
+ \((2g - 1)\)-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 Lickrish generators \(\tau_{\mu_1}, \ldots, \tau_{\mu_g}, \tau_{\alpha_1},
+\ldots, \tau_{\alpha_g} \in \Mod(S_g^b)\) on \(H_1(S_g, \mathbb{C}) \cong
+\mathbb{C}^{2g}\) -- represented in the standard basis \([\mu_1], \ldots,
+[\mu_g], [\alpha_1], \ldots, [\alpha_g]\) for \(H_1(S_g, \mathbb{C})\).
+\begin{align*}
+ (\tau_{\mu_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_{\alpha_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 embeding \(B_3^g\) in \(\Mod(S_g^b)\) via
+\begin{align*}
+ B_3^g & \to \Mod(S_g^b) \\
+ a_i & \mapsto \tau_{\mu_i} \\
+ b_i & \mapsto \tau_{\alpha_i}
+\end{align*}
+we can see that any \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\) in a
+certain class of representation satisfying some technical conditions must be
+conjugate to the symplectic representation \(\Mod(S_g^b) \to
+\operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\).
+
+Korkmaz then goes on to show that such technical conditions are met for any
+nontrivial \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\). 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 agrees with the action of \(\Mod(S_g^b)\) on
+\(H_1(S_g, \mathbb{C})\), concluding the classification of \(2g\)-dimensional
+representations.
+
+% TODO: Add some final comments about how the rest of the landscape of
+% representations is generally unknown and how there is a lot to study in here
+Recently, Kasahara also classified the \((2g+1)\)-dimensional representations
+of \(\Mod(S_g^b)\) for \(g \ge 7\) in terms of certain twisted \(1\)-cohomology
+groups \cite{kasahara}.