diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -22,6 +22,7 @@
\end{example}
% TODO: Is every quotient of a weight module a weight module too?
+% TODO: I think so!
\begin{example}
Proposition~\ref{thm:verma-is-weight-mod} and
proposition~\ref{thm:max-verma-submod-is-weight} imply that the Verma module
@@ -120,6 +121,10 @@
\end{enumerate}
\end{definition}
+% TODOO: Add a discussion on how this may sound unintuitive, but the motivation
+% comes from the relationship between highest weight modules and coherent
+% families
+
\begin{definition}
A coherent family \(\mathcal{M}\) called \emph{irreducible} if
\(\mathcal{M}_\lambda\) is a simple
@@ -133,7 +138,27 @@
\(V\) as a subquotient.
\end{definition}
-% TODO: Define the semisimplification of a coherent family
+% Mathieu's proof of this is somewhat profane, I don't think it's worth
+% including it in here
+\begin{proposition}
+ Any coherent family \(\mathcal{M}\) has finite length as a
+ \(\mathfrak{g}\)-module.
+\end{proposition}
+
+% TODO: Add a proof!
+\begin{corollary}
+ Given a coherent family \(\mathcal{M}\) and a composition series \(0 =
+ \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_n =
+ \mathcal{M}\), the \(\mathfrak{g}\)-module
+ \[
+ \mathcal{M}^{\operatorname{ss}}
+ = \bigoplus_i \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
+ \]
+ is also a coherent family, called \emph{the semisimplification\footnote{This
+ name is due to the fact that $\mathcal{M}^{\operatorname{ss}}$ is the direct
+ sum of irreducible $\mathfrak{g}$-modules} of \(\mathcal{M}\)}.
+\end{corollary}
+
\begin{theorem}[Mathieu]
Let \(V\) be an infinite-dimensional admissible irreducible
\(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple
@@ -161,7 +186,7 @@
\[
\operatorname{Ext}(V)
\cong \mathcal{M}^{\operatorname{ss}}
- = \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j},
+ = \bigoplus_i \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i},
\]
so that \(V\) is contained in \(\operatorname{Ext}(V)\).