memoire-m2

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