memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 9a5f978aa9abc294c25f5d086bbc7d10f086a059
Parent: e221d5fd6496104788fee9af632808f7690cfdd9
Author: Pablo <pablo-pie@riseup.net>
Date: Tue, 18 Jun 2024 14:45:16 +0000

Relabeled the Lickorish generators

Also redrew the labels for the Humphreays generators

Diffstat

6 files changed, 360 insertions, 335 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	images/humphreys-gens.svg	144	61	83
Modified	images/lickorish-gens-gen-2.svg	66	40	26
Modified	images/lickorish-gens-korkmaz-proof.svg	123	70	53
Modified	images/lickorish-gens.svg	118	65	53
Modified	sections/representations.tex	242	123	119
Modified	sections/twists.tex	2	1	1

diff --git a/images/humphreys-gens.svg b/images/humphreys-gens.svg
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diff --git a/sections/representations.tex b/sections/representations.tex
@@ -35,7 +35,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
 \begin{proof}[Sketch of proof]
   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 \(\alpha_1, \alpha_2, \mu_1, \mu_2,
+  \lambda v \}\) its eigenspaces. Let \(\alpha_1, \alpha_2, \beta_1, \beta_2,
   \gamma, \eta_1, \ldots, \eta_{b - 1} \subset S_2^b\) be the curves of the
   Lickorish generators from Theorem~\ref{thm:lickorish-gens}, as shown in
   Figure~\ref{fig:lickorish-gens-genus-2}.
@@ -49,13 +49,14 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
   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 \(\Mod(S_2^b)' \subset \ker \rho\) -- i.e. \(\rho(\Mod(S_2^b))\) is
-  Abelian. Given the braid relation
+  = L_{\beta_1}\), so that \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker
+  \rho\) 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},
+    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 \(L_{\mu_1}\) commute.
+  this amounts to showing \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute.
 
   To that end, we exhausively analyse all of the possible Jordan forms
   \begin{align*}
@@ -119,74 +120,75 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
     \end{pmatrix}
     & \quad{\normalfont(9)}
   \end{align*}
-  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 \(E_{\mu_2 =
-  \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\).
+  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 (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 \(L_{\mu_1}\) lie in some Abelian subgroup of \(\GL_n(\mathbb{C})\) --
+  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 \(\alpha_2\). 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
+  For cases (8) and (9) 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 \(\Mod(S_2^b)\)-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]
+    f \cdot [\alpha_2]      & = [\alpha_1]
     &
-    g \cdot [\mu_2]      & = [\alpha_1]
+    g \cdot [\alpha_2]      & = [\beta_1]
     &
-    h_i \cdot [\mu_2]    & = [\mu_2]    \\
-    f \cdot [\alpha_2]   & = [\alpha_1]
+    h_i \cdot [\alpha_2]    & = [\alpha_2]    \\
+    f \cdot [\beta_2]   & = [\beta_1]
     &
-    g \cdot [\alpha_2]   & = [\gamma]
+    g \cdot [\beta_2]   & = [\gamma]
     &
-    h_i \cdot [\alpha_2] & = [\eta_i].
+    h_i \cdot [\beta_2] & = [\eta_i].
   \end{align*}
   In particular,
   \begin{align*}
-    f   \tau_{\mu_2}    f^{-1}   & = \tau_{\mu_1}
+    f   \tau_{\alpha_2}    f^{-1}   & = \tau_{\alpha_1}
     &
-    g   \tau_{\mu_2}    g^{-1}   & = \tau_{\alpha_1}
+    g   \tau_{\alpha_2}    g^{-1}   & = \tau_{\beta_1}
     &
-    h_i \tau_{\mu_2}    h_i^{-1} & = \tau_{\mu_2}     \\
-    f   \tau_{\alpha_2} f^{-1}   & = \tau_{\alpha_1}
+    h_i \tau_{\alpha_2}    h_i^{-1} & = \tau_{\alpha_2}     \\
+    f   \tau_{\beta_2} f^{-1}   & = \tau_{\beta_1}
     &
-    g   \tau_{\alpha_2} g^{-1}   & = \tau_{\gamma}
+    g   \tau_{\beta_2} g^{-1}   & = \tau_{\gamma}
     &
-    h_i \tau_{\alpha_2} h_i^{-1} & = \tau_{\eta_i}.
+    h_i \tau_{\beta_2} h_i^{-1} & = \tau_{\eta_i}.
   \end{align*}
   and thus
   \begin{align*}
-    E_{\mu_1 = \lambda}
-    = \rho(f) E_{\mu_2 = \lambda}
-    & = \rho(f) E_{\alpha_2 = \lambda}
-    = E_{\alpha_1 = \lambda}
-    \\
     E_{\alpha_1 = \lambda}
-    = \rho(g) E_{\mu_2 = \lambda}
-    & = \rho(g) E_{\alpha_2 = \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_{\mu_2 = \lambda}
-    & = \rho(h_i) E_{\alpha_2 = \lambda}
-    = E_{\alpha_2 = \lambda}.
+    = \rho(h_i) E_{\alpha_2 = \lambda}
+    & = \rho(h_i) E_{\beta_2 = \lambda}
+    = E_{\beta_2 = \lambda}.
   \end{align*}
   In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} =
-  E_{\mu_1 = \lambda} = E_{\mu_2 = \lambda} = E_{\gamma = \lambda} =
-  E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b - 1} = \lambda}\) is
-  invariant under the action of all Lickorish generators.
+  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 \(\bar \rho : \Mod(S_2^b) \to
-  \GL(E_{\mu_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\mu_2 =
+  \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\alpha_2 =
   \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By case (2), \(\bar
   \rho(f) = 1\) for all \(f \in \Mod(S_2^b)'\), given that \(\bar
   \rho(\Mod(S_2^b))\) is Abelian. Thus
@@ -203,28 +205,29 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
   get \(\rho(\Mod(S_2^b)') = 1\): any homomorphism from a perfect group to a
   solvable group is trivial.
 
-  Finally, if \(E_{\mu_2 = \lambda} \ne E_{\alpha_2 = \lambda}\) and
-  the Jordan form of \(L_{\mu_2}\) is given by (8) then
+  Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and
+  the Jordan form of \(L_{\alpha_2}\) is given by (8) then
   \[
     0
-    \subsetneq E_{\mu_2 = \lambda} \cap E_{\alpha_2 = \lambda}
-    \subsetneq E_{\mu_2 = \lambda}
+    \subsetneq E_{\alpha_2 = \lambda} \cap E_{\beta_2 = \lambda}
+    \subsetneq E_{\alpha_2 = \lambda}
     \subsetneq V
   \]
-  is a flag of subspaces invariant under \(L_{\mu_1}\) 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 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 \(E_{\mu_2 = \lambda}
-  \ne E_{\alpha_2 = \lambda}\) we use (\ref{eq:braid-rel-induction-basis})
-  to conclude \(L_{\mu_1}\) and \(L_{\alpha_1}\) commute -- again, see
+  is a flag of subspaces invariant under \(L_{\alpha_1}\) and \(L_{\beta_1}\),
+  for \(\alpha_2\) \(\beta_2\) are disjoint from \(\alpha_1 \cup \beta_1\) and
+  thus \([\tau_{\alpha_2}, \tau_{\alpha_1}] = [\tau_{\alpha_2}, \tau_{\beta_1}]
+  = [\tau_{\beta_2}, \tau_{\alpha_1}] = [\tau_{\beta_2}, \tau_{\beta_1}] = 1\).
+  In this case we can find a basis for \(\mathbb{C}^3\) with respect to wich
+  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 \(v_1 \in E_{\alpha_2 = \lambda} \cap E_{\beta_2 =
+  \lambda}\), \(v_2 \in V_{L_{\alpha_2}}\) 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 \(L_{\alpha_2}\) 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}
 
@@ -241,14 +244,15 @@ representations.
   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
+  Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots,
+  \gamma_{g - 1}, \eta_1, \ldots, \eta_{b - 1} \subset S_g^b\) be the curves
   from the Lickorish generators of \(\Mod(S_g^b)\), as in
-  Theorem~\ref{thm:lickorish-gens}. Once again, let \(L_\alpha =
+  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 \(R \cong S_{g -
   1}^1\) 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}
@@ -286,28 +290,28 @@ representations.
   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
+  \tau_{\beta_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
+  generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), 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 \(W =
-  \bigoplus_i E_{\mu_g = \lambda_i}\) of the corresponding eigenspaces has
+  \(L_{\alpha_g}\). 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 \(\mu_g\) lies outside of \(R\), each \(E_{\mu_g =
+  observe that since \(\alpha_g\) lies outside of \(R\), each \(E_{\alpha_g =
   \lambda_i}\) 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
+  \(\tau_{\alpha_1}, \ldots, \tau_{\alpha_{g - 1}}, \tau_{\beta_1}, \ldots,
+  \tau_{\beta_{g - 1}}\), \(\tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - 2}}\) of
+  \(R \cong S_{g - 1}^1\) all commute with \(\tau_{\alpha_g}\) and thus preserve
   the eigenspaces of its action on \(\mathbb{C}^n\).
 
-  If no sum of the form \(\bigoplus_i E_{\mu_g = \lambda_i}\) 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
-  be one of
+  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
+  eigenvalues \(\lambda\) of \(L_{\alpha_g}\). Hence the Jordan form of
+  \(L_{\alpha_g}\) has to be one of
   \begin{align*}
     \begin{pmatrix}
       \lambda & 0       & 0      & \cdots & 0       & 0       \\
@@ -349,49 +353,49 @@ representations.
   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 \(\mathbb{C}^n\) 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
+  \(L_{\alpha_i}, L_{\beta_i}, L_{\gamma_i},  L_{\eta_i}\) is conjugate to
+  \(L_{\alpha_g} = \lambda\), so all Lickorish generators of \(\Mod(S_g^b)\)
+  act 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), \(W =
+  \ker (L_{\alpha_g} - \lambda)^2\) is a \(2\)-dimensional
+  \(\Mod(R)\)-invariant subspace.
+
+  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 \(f_i, g_i, h_i \in
   \Mod(S_g^b)\) with
   \begin{align*}
-    f_i \tau_{\mu_g}    f_i^{-1} & = \tau_{\mu_i}
+    f_i \tau_{\alpha_g}    f_i^{-1} & = \tau_{\alpha_i}
     &
-    g_i \tau_{\mu_g}    g_i^{-1} & = \tau_{\alpha_i}
+    g_i \tau_{\alpha_g}    g_i^{-1} & = \tau_{\beta_i}
     &
-    h_i \tau_{\mu_g}    h_i^{-1} & = \tau_{\mu_g}
+    h_i \tau_{\alpha_g}    h_i^{-1} & = \tau_{\alpha_g}
     \\
-    f_i \tau_{\alpha_g} f_i^{-1} & = \tau_{\alpha_i}
+    f_i \tau_{\beta_g} f_i^{-1} & = \tau_{\beta_i}
     &
-    g_i \tau_{\alpha_g} g_i^{-1} & = \tau_{\gamma_i}
+    g_i \tau_{\beta_g} g_i^{-1} & = \tau_{\gamma_i}
     &
-    h_i \tau_{\alpha_g} h_i^{-1} & = \tau_{\eta_i}.
+    h_i \tau_{\beta_g} h_i^{-1} & = \tau_{\eta_i}.
   \end{align*}
   and thus
   \[
-    E_{\mu_1 = \lambda} = \cdots = E_{\mu_g = \lambda}
-    = E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda}
+    E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda}
+    = E_{\beta_1 = \lambda} = \cdots = E_{\beta_g = \lambda}
     = E_{\gamma_1 = \lambda} = \cdots = E_{\gamma_{g - 1} = \lambda}
     = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b - 1} = \lambda}.
   \]
 
   In particular, we can find a basis for \(\mathbb{C}^n\) with respect to which
-  the matrix of any Lickorish generators has the form
+  the matrix of any Lickorish generator has the form
   \[
     \begin{pmatrix}
       \lambda & 0       & \cdots & 0       & *      \\
@@ -408,7 +412,7 @@ representations.
   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}
+  \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}
@@ -477,14 +481,14 @@ to as \emph{the main lemma}. Namely\dots
 \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 \(H_1(S_g, \mathbb{C})\).
+that for \(n = g\) and \(m = 2g\) the matrices in Lemma~\ref{thm:main-lemma}
+coincide with the action of the Lickrish generators \(\tau_{\alpha_1}, \ldots,
+\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(S_g^b)\) on
+\(H_1(S_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard
+basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for
+\(H_1(S_g, \mathbb{C})\).
 \begin{align*}
-  (\tau_{\mu_i})_* & =
+  (\tau_{\alpha_i})_* & =
   \left(
   \begin{array}{c|c|c}
     1 & 0                                            & 0 \\ \hline
@@ -492,7 +496,7 @@ the Lickrish generators \(\tau_{\mu_1}, \ldots, \tau_{\mu_g}, \tau_{\alpha_1},
     0 & 0                                            & 1
   \end{array}
   \right) &
-  (\tau_{\alpha_i})_* & =
+  (\tau_{\beta_i})_* & =
   \left(
   \begin{array}{c|c|c}
     1 & 0                                             & 0 \\ \hline
@@ -505,8 +509,8 @@ the Lickrish generators \(\tau_{\mu_1}, \ldots, \tau_{\mu_g}, \tau_{\alpha_1},
 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}
+  a_i   & \mapsto \tau_{\alpha_i}    \\
+  b_i   & \mapsto \tau_{\beta_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

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -451,7 +451,7 @@ generators}.
 
 \begin{theorem}[Lickorish generators]\label{thm:lickorish-gens}
   If \(g \ge 1\) then \(\Mod(S_g^b)\) is generated by the Dehn twists about the
-  curves \(\alpha_1, \ldots, \alpha_g, \mu_1, \ldots, \mu_g, \gamma_1,
+  curves \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1,
   \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{b - 1}\) as in
   Figure~\ref{fig:lickorish-gens}
 \end{theorem}