lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -527,15 +527,8 @@ follows.
 \begin{definition}
   A coherent family \(\mathcal{M}\) is called \emph{simple} if it contains no
   proper coherent subfamilies -- i.e. \(\mathcal{M}\) is a simple object in the
-  full subcategory of coherent families. Equivalently\footnote{It is easy to
-  see that if $\mathcal{M}_\lambda$ is a simple
-  $\mathcal{U}(\mathfrak{g})_0$-module for some $\lambda \in \mathfrak{h}^*$
-  then $\mathcal{M}$ is a simple object in the category of coherent families,
-  for if $\mathcal{N} \subset \mathcal{M}$ is a nonzero coherent subfamily
-  $\mathcal{N}_\lambda = \mathcal{M}_\lambda$ and therefore $\deg \mathcal{N} =
-  \deg \mathcal{M}$, which implies $\mathcal{N} = \mathcal{M}$. The converse is
-  also true, but its proof was deemed too technical to be included in here.},
-  we call \(\mathcal{M}\) simple if \(\mathcal{M}_\lambda\) is a simple
+  full subcategory of coherent families. Equivalently, we call \(\mathcal{M}\)
+  simple if \(\mathcal{M}_\lambda\) is a simple
   \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
   \mathfrak{h}^*\).
 \end{definition}
@@ -556,8 +549,6 @@ to a completely reducible coherent extension of \(V\).
   \(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
 \end{lemma}
 
-% TODOO: Point out this construction is NOT functorial, since it depends on the
-% choice of composition series
 \begin{corollary}
   Let \(\mathcal{M}\) be a coherent family of degree \(d\). There exists a
   unique completely reducible coherent family
@@ -568,7 +559,13 @@ to a completely reducible coherent extension of \(V\).
 
   Namely, if \(\lambda \in \mathfrak{h}^*\) and \(0 = \mathcal{M}_{\lambda 0}
   \subset \mathcal{M}_{\lambda 1} \subset \cdots \subset \mathcal{M}_{\lambda
-  n_\lambda} = \mathcal{M}[\lambda]\) is a composition series,
+  n_\lambda} = \mathcal{M}[\lambda]\) is a composition series\footnote{Notice
+  that $\mathcal{M}[\lambda] = \mathcal{N}[\mu]$ for any $\mu \in \lambda + Q$.
+  Hence the sum $\bigoplus_{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q}}
+  \bigoplus_i \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}$ is
+  independant of the choice of representative for $\lambda + Q$ -- at least as
+  long as we choose $\mathcal{M}_{\lambda i} = \mathcal{M}_{\mu i}$ for all
+  $\mu \in \lambda + Q$ and $i$.},
   \[
     \mathcal{M}^{\operatorname{ss}}
     \cong \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}}
@@ -636,6 +633,20 @@ to a completely reducible coherent extension of \(V\).
   is polynomial in \(\mu \in \mathfrak{h}^*\).
 \end{proof}
 
+\begin{note}
+  Althought we have provided an explicit construction of
+  \(\mathcal{M}^{\operatorname{ss}}\) in terms of \(\mathcal{M}\), we should
+  point out this construction is not fuctorial. First, given an intertwiner \(T
+  : \mathcal{M} \to \mathcal{N}\) between coherent families, it is unclear what
+  \(T^{\operatorname{ss}} : \mathcal{M}^{\operatorname{ss}} \to
+  \mathcal{N}^{\operatorname{ss}}\) is supposed to be. Secondly, and this is
+  more relevant, our construction depends on the choice of composition series
+  \(0 = \mathcal{M}_{\lambda 0} \subset \cdots \subset \mathcal{M}_{\lambda
+  n_\lambda} = \mathcal{M}[\lambda]\). While different choices of composition
+  series yield isomorphic results, there is no canonical isomorphism.
+  In addition, there is no canonical choice of composition series.
+\end{note}
+
 As promised, if \(\mathcal{M}\) is a coherent extension of \(V\) then so is
 \(\mathcal{M}^{\operatorname{ss}}\).
 
@@ -1290,10 +1301,12 @@ It should now be obvious\dots
   There exists a coherent extension \(\mathcal{M}\) of \(V\).
 \end{proposition}
 
-% TODOO: Point out that here we have to fix representatives of the cosets, so
-% this construction is, once again, not functorial
 \begin{proof}
-  Take
+  Take\footnote{Here we fix some $\lambda_t \in t$ for each $Q$-coset $t \in
+  \mfrac{\mathfrak{h}^*}{Q}$. While there is a natural isomorphism
+  $\theta_\lambda \Sigma^{-1} V \isoto \theta_\mu \Sigma^{-1} V$ for each $\mu
+  \in \lambda + Q$, they are not the same representation strictly speaking.
+  This is yet another obstruction to the functoriality of our constructions.}
   \[
     \mathcal{M}
     = \bigoplus_{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q}}