lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -146,6 +146,14 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   m(\lambda)_n\).
 \end{proposition}
 
+\begin{definition}
+  We denote by \(\mathscr{Q}\) the set of the \(m \in (\sfrac{1}{2} +
+  \mathbb{Z})^n\) such that \(m_1 > m_2 > \cdots > m_{n - 1} > \pm m_n\). We
+  also consider the canonical partition \(\mathscr{Q} = \mathscr{Q}^+ \cup
+  \mathscr{Q}^-\) where \(\mathscr{Q}^+ = \{ m \in \mathscr{Q} : m_n > 0 \}\)
+  and \(\mathscr{Q}^- = \{ m \in \mathscr{Q} : m_n < 0\}\).
+\end{definition}
+
 % TODO: Prove this
 % Given the Harish-Chandra theorem and the previous proposition, all its left
 % is to show that of m(λ)_n = - m(μ)_n then indeed Ext(L(λ)) = Ext(L(μ)). I
@@ -156,8 +164,7 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   \mExt(L(\mu))\) if, and only if \(m(\lambda)_i = m(\mu)_i\) for \(i < n\) and
   \(m(\lambda)_n = \pm m(\mu)_n\). In particular, the isomorphism classes of
   semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-families are
-  parameterized by the \(n\)-tuples \(m \in (\sfrac{1}{2} + \mathbb{Z})^n\)
-  with \(m_1 > m_2 > \cdots > m_n > 0\).
+  parameterized by \(\mathscr{Q}^+\).
 \end{theorem}
 
 \section{Coherent \(\mathfrak{sl}_n(K)\)-families}
@@ -229,14 +236,23 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
 
 % TODO: Change the notation for 𝓟 ? The paper on affine vertex algebras calls
 % this graph 𝓑
+% TODO: Explain the significance of 𝓟 + and 𝓟 -: these are the subsets whose
+% union corresponds to condition (i)
+\begin{definition}
+  We denote by \(\mathscr{P}\) the set of \(\mathfrak{sl}_n\)-sequences \(m\)
+  such that \(m_i - m_{i + 1}\) is a positive integer for all but a single \(i
+  < n\). We also consider the subsets \(\mathscr{P}^+ = \{m \in \mathscr{P} :
+  m_1 - m_2 \ \text{is \emph{not} a positive integer}\}\) and \(\mathscr{P}^- =
+  \{m \in \mathscr{P} : m_{n-1} - m_n \ \text{is \emph{not} a positive
+  integer}\}\).
+\end{definition}
+
 % TODO: Explain the intuition behind defining the arrows like so: the point is
 % that if there is an arrow m(λ) → m(μ) then μ = σ_i ∙ λ for some i, which
 % 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 \(\mathscr{P}\) the set of \(\mathfrak{sl}_n\)-sequences \(m\) such
-  that \(m_i - m_{i + 1}\) is a positive integer for all but a single \(i <
-  n\). Given \(m, m' \in \mathscr{P}\), say there is an arrow \(m \to m'\) if
+  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}
@@ -251,12 +267,6 @@ It should then be obvious that\dots
   \mExt(L(\lambda))\).
 \end{proposition}
 
-\begin{definition}
-  Let \(\mathscr{P}^+ = \{m \in \mathscr{P} : m_1 - m_2 \ \text{is \emph{not} a
-  positive integer}\}\) and \(\mathscr{P}^- = \{m \in \mathscr{P} : m_{n-1} -
-  m_n \ \text{is \emph{not} a positive integer}\}\).
-\end{definition}
-
 A weight \(\lambda \in \mathfrak{h}^*\) is called \emph{regular} if \((\lambda
 + \rho)(H_\alpha) \ne 0\) for all \(\alpha \in \Delta\). In terms of
 \(\mathfrak{sl}_n\)-sequences, \(\lambda\) is regular if, and only if
@@ -360,7 +370,8 @@ all \(i\) and \(j\).
   \(m(\mu)\) lie in the same connected component of \(\mathscr{P}\). In
   particular, the isomorphism classes of semisimple irreducible coherent
   \(\mathfrak{sl}_n(K)\)-families are parameterized by the set
-  \(\pi_0(\mathscr{P})\) of the connected components of \(\mathscr{P}\).
+  \(\pi_0(\mathscr{P})\) of the connected components of \(\mathscr{P}\), as
+  well as by \(\mathscr{P}^+\).
 \end{theorem}
 
 % TODO: Change this