diff --git a/sections/representations.tex b/sections/representations.tex
@@ -60,13 +60,13 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
\begin{align*}
\begin{pmatrix}
\lambda & 0 \\
- 0 & \mu
+ 0 & \lambda
\end{pmatrix}
& \quad{\normalfont(1)}
&
\begin{pmatrix}
\lambda & 0 \\
- 0 & \lambda
+ 0 & \mu
\end{pmatrix}
& \quad{\normalfont(2)}
&
@@ -77,31 +77,31 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
& \quad{\normalfont(3)}
\\
\begin{pmatrix}
- \lambda & 0 & 0 \\
- 0 & \mu & 0 \\
- 0 & 0 & \nu
- \end{pmatrix}
- & \quad{\normalfont(4)}
- &
- \begin{pmatrix}
\lambda & 0 & 0 \\
0 & \lambda & 0 \\
0 & 0 & \lambda
\end{pmatrix}
- & \quad{\normalfont(5)}
+ & \quad{\normalfont(4)}
&
\begin{pmatrix}
\lambda & 0 & 0 \\
- 0 & \mu & 1 \\
- 0 & 0 & \mu
+ 0 & \mu & 0 \\
+ 0 & 0 & \nu
\end{pmatrix}
- & \quad{\normalfont(6)}
- \\
+ & \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}
@@ -123,11 +123,39 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
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 (7) we use the change of coordinates principle and the braid
- relation (\ref{eq:braid-rel-induction-basis}) to show that \(L_{\alpha_1}\)
- and \(L_{\beta_1}\) 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 \(\beta_2\). In these cases, the
+ 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 \(\GL_n(\mathbb{C})\) and thus commute.
+
+ \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.
+
+ \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
@@ -306,8 +334,8 @@ representations.
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 \(m - 2\) there must be at most \(2\) distinct
- eigenvalues and \(\dim E_{\alpha_g = \lambda} = 1, m - 1, m\) for all
+ lying between \(2\) and \(n - 2\) there must be at most \(2\) distinct
+ eigenvalues and \(\dim E_{\alpha_g = \lambda} = 1, n - 1, n\) for all
eigenvalues \(\lambda\) of \(L_{\alpha_g}\). Hence the Jordan form of
\(L_{\alpha_g}\) has to be one of
\begin{align*}
@@ -347,29 +375,35 @@ representations.
\end{pmatrix}
& \quad{\normalfont(4)}
\end{align*}
- for \(\lambda \ne \mu\). We analyze each one of these sporadic cases
+ for \(\lambda \ne \mu\). We analyze the first two sporadic cases
individually.
- For case (1), 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 \(\Mod(S_g^p)\)
- act on \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well.
- Hence \(\rho(\Mod(S_g^p))\) is cyclic and thus Abelian. In case (2), \(W =
- \ker (L_{\alpha_g} - \lambda)^2\) is a \(2\)-dimensional
- \(\Mod(R)\)-invariant subspace.
+ \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 \(\Mod(S_g^p)\) act on
+ \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence
+ \(\rho(\Mod(S_g^p)) = \langle \lambda \rangle\) is Abelian.
+
+ \item[\bfseries\color{highlight}(2)]
+ In this case, \(W = \ker (L_{\alpha_g} - \lambda)^2\) is a
+ \(2\)-dimensional \(\Mod(R)\)-invariant subspace.
+ \end{enumerate}
- For cases (3) and (4) we consider two situations: \(E_{\alpha_g = \lambda}
+ For cases (3) and (4), we consider two situations: \(E_{\alpha_g = \lambda}
\ne E_{\beta_g = \lambda}\) or \(E_{\alpha_g = \lambda} = E_{\beta_g =
- \lambda}\). In the first case, \(W = E_{\alpha_g = \lambda} \cap E_{\beta_g =
- \lambda}\) is a \((m - 2)\)-dimensional \(\Mod(R)\)-invariant subspace: since
- \(L_{\alpha_g}\) and \(L_{\beta_g}\) are conjugate and \(\beta_g\) lies
- outside of \(R\), both \(E_{\alpha_g = \lambda}\) and \(E_{\beta_g =
- \lambda}\) are \(\Mod(R)\)-invariant \((m - 1)\)-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 the
- change of coordinates principle that there are \(f_i, g_i, h_i \in
+ \lambda}\). 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 \(\Mod(R)\)-invariant subspace: since \(L_{\alpha_g}\) and
+ \(L_{\beta_g}\) are conjugate and \(\beta_g\) lies outside of \(R\), both
+ \(E_{\alpha_g = \lambda}\) and \(E_{\beta_g = \lambda}\) are
+ \(\Mod(R)\)-invariant \((m - 1)\)-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
+ the change of coordinates principle that there are \(f_i, g_i, h_i \in
\Mod(S_g^p)\) with
\begin{align*}
f_i \tau_{\alpha_g} f_i^{-1} & = \tau_{\alpha_i}
@@ -392,8 +426,8 @@ representations.
= E_{\eta_1 = \lambda} = \cdots = E_{\eta_{p-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
+ In particular, we can find a basis for \(\mathbb{C}^n\) with respect to
+ which the matrix of any Lickorish generator has the form
\[
\begin{pmatrix}
\lambda & 0 & \cdots & 0 & * \\
@@ -403,9 +437,9 @@ representations.
0 & 0 & \cdots & 0 & *
\end{pmatrix}.
\]
- Since the group of upper triangular matrices is solvable and \(\Mod(S_g^p)\)
- is perfect, it follows that \(\rho(\Mod(S_g^p))\) is trivial. This concludes
- the proof \(\rho(\Mod(S_g^p))\) is Abelian.
+ Since the group of upper triangular matrices is solvable and
+ \(\Mod(S_g^p)\) is perfect, it follows that \(\rho(\Mod(S_g^p))\) is
+ trivial. This concludes the proof \(\rho(\Mod(S_g^p))\) is Abelian.
To see that \(\rho(\Mod(S_g^p)) = 1\) for \(g \ge 3\) we note that, since
\(\rho(\Mod(S_g^p))\) is Abelian, \(\rho\) factors though the Abelinization