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}}