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