lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -239,9 +239,9 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
 % implies L(μ) is contained in 𝓔𝔁𝓽(L(λ)) - so that L(μ) is also bounded and
 % 𝓔𝔁𝓽(L(λ)) ≅ 𝓔𝔁𝓽(L(μ))
 \begin{definition}
-  Denote by \(\mathcal{P}\) the set of \(\mathfrak{sl}_{n + 1}(K)\)-sequences
+  Denote by \(\mathscr{P}\) the set of \(\mathfrak{sl}_{n + 1}(K)\)-sequences
   \(m\) such that \(m_i - m_{i + 1}\) is a positive integer for all but a
-  single \(i \le n\). Given \(m, m' \in \mathcal{P}\), say there is an arrow
+  single \(i \le n\). Given \(m, m' \in \mathscr{P}\), say there is an arrow
   \(m \to m'\) if the unique \(i\) such that \(m_i - m_{i + 1}\) is not a
   positive integer is such that \(m' = \sigma_i \cdot m\).
 \end{definition}
@@ -250,15 +250,15 @@ It should then be obvious that\dots
 
 \begin{proposition}
   Let \(\lambda \notin P^+\) be such that \(L(\lambda)\) is bounded -- so that
-  \(m(\lambda) \in \mathcal{P}\) -- and suppose that \(\mu \in \mathfrak{h}^*\)
-  is such that \(m(\mu) \in \mathcal{P}\) and there is an arrow \(m(\lambda)
+  \(m(\lambda) \in \mathscr{P}\) -- and suppose that \(\mu \in \mathfrak{h}^*\)
+  is such that \(m(\mu) \in \mathscr{P}\) and there is an arrow \(m(\lambda)
   \to m(\mu)\). Then \(L(\mu)\) is also bounded and \(\mExt(L(\mu)) \cong
   \mExt(L(\lambda))\).
 \end{proposition}
 
 \begin{definition}
-  Let \(\mathcal{P}^+ = \{m \in \mathcal{P} : m_1 - m_2 \ \text{is not a
-  positive integer}\}\) and \(\mathcal{P}^- = \{m \in \mathcal{P} : m_n - m_{n
+  Let \(\mathscr{P}^+ = \{m \in \mathscr{P} : m_1 - m_2 \ \text{is not a
+  positive integer}\}\) and \(\mathscr{P}^- = \{m \in \mathscr{P} : m_n - m_{n
   + 1} \ \text{is not a positive integer}\}\).
 \end{definition}
 
@@ -283,7 +283,7 @@ It should then be obvious that\dots
 % theorem? Perhaps it's best to create another lemma for this
 % TODOO: Define the notation for σ_i beforehand
 \begin{proposition}
-  The connected component of some \(m \in \mathcal{P}\) is given by the
+  The connected component of some \(m \in \mathscr{P}\) is given by the
   following.
   \begin{enumerate}
     \item If \(m\) is regular and integral then there exists\footnote{Notice
@@ -308,8 +308,8 @@ It should then be obvious that\dots
         \end{tikzcd}
       \]
       for some unique \(i\), with \(\sigma_1 \cdots \sigma_i \cdot m' \in
-      \mathcal{P}^+\) and \(\sigma_n \cdots \sigma_i \cdot m' \in
-      \mathcal{P}^-\).
+      \mathscr{P}^+\) and \(\sigma_n \cdots \sigma_i \cdot m' \in
+      \mathscr{P}^-\).
 
     \item If \(m\) is singular then there exists unique \(m' \in W \cdot m\)
       and \(i\) such that \(m_1' > m_2' > \cdots > m_i' = m_{i + 1}' > \cdots >
@@ -331,8 +331,8 @@ It should then be obvious that\dots
           \sigma_n \cdots \sigma_{i+1} \cdot m'                \lar &
         \end{tikzcd}
       \]
-      with \(\sigma_1 \cdots \sigma_{i-1} \cdot m' \in \mathcal{P}^+\) and
-      \(\sigma_n \cdots \sigma_{i+1} \cdot m' \in \mathcal{P}^-\).
+      with \(\sigma_1 \cdots \sigma_{i-1} \cdot m' \in \mathscr{P}^+\) and
+      \(\sigma_n \cdots \sigma_{i+1} \cdot m' \in \mathscr{P}^-\).
 
     \item If \(m\) is non-integral then there exists unique \(m' \in W \cdot
       m\) such that \(m_2' > m_3' > \cdots > m_{n + 1}'\), in which case the
@@ -346,8 +346,8 @@ It should then be obvious that\dots
           \sigma_n \cdots \sigma_1 \cdot m'      \lar &
         \end{tikzcd}
       \]
-      with \(m' \in \mathcal{P}^+\) and \(\sigma_n \cdots \sigma_1 \cdot m' \in
-      \mathcal{P}^-\).
+      with \(m' \in \mathscr{P}^+\) and \(\sigma_n \cdots \sigma_1 \cdot m' \in
+      \mathscr{P}^-\).
   \end{enumerate}
 \end{proposition}
 
@@ -363,10 +363,10 @@ It should then be obvious that\dots
 \begin{theorem}[Mathieu]
   Given \(\lambda, \mu \notin P^+\) with \(L(\lambda)\) and \(L(\mu)\) bounded,
   \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and
-  \(m(\mu)\) lie in the same connected component of \(\mathcal{P}\). In
+  \(m(\mu)\) lie in the same connected component of \(\mathscr{P}\). In
   particular, the isomorphism classes of semisimple irreducible coherent
   \(\mathfrak{sl}_{n + 1}(K)\)-families are parameterized by the set
-  \(\pi_0(\mathcal{P})\) of the connected components of \(\mathcal{P}\).
+  \(\pi_0(\mathscr{P})\) of the connected components of \(\mathscr{P}\).
 \end{theorem}
 
 % TODO: Change this