diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -185,15 +185,14 @@
% composition series are conjugate
\begin{corollary}
Let \(\{\lambda_i\}_i\) be a set of representatives of the \(Q\)-cosets of
- \(\mathfrak{h}^*\).
- Given a coherent family \(\mathcal{M}\) of degree \(d\) and composition
- series \(0 = \mathcal{M}_0[\lambda_i] \subset \mathcal{M}_1[\lambda_i]
- \subset \cdots \subset \mathcal{M}_n[\lambda_i] = \mathcal{M}[\lambda_i]\),
- the \(\mathfrak{g}\)-module
+ \(\mathfrak{h}^*\). Given a coherent family \(\mathcal{M}\) of degree \(d\)
+ and composition series \(0 = \mathcal{M}_{i 0} \subset \mathcal{M}_{i 1}
+ \subset \cdots \subset \mathcal{M}_{i n_i} = \mathcal{M}[\lambda_i]\), the
+ \(\mathfrak{g}\)-module
\[
\mathcal{M}^{\operatorname{ss}}
= \bigoplus_{i j}
- \mfrac{\mathcal{M}_{j + 1}[\lambda_i]}{\mathcal{M}_j[\lambda_i]}
+ \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
\]
is also a coherent family of degree \(d\), called \emph{the
semisimplification\footnote{This name is due to the fact that
@@ -202,29 +201,33 @@
\end{corollary}
\begin{proof}
- We know form examples~\ref{ex:submod-is-weight-mod} and
- \ref{ex:quotient-is-weight-mod} that each quotient \(\mfrac{\mathcal{M}_{j +
- 1}[\lambda_i]}{\mathcal{M}_j[\lambda_i]}\) is weight module. Hence
+ We know from examples~\ref{ex:submod-is-weight-mod} and
+ \ref{ex:quotient-is-weight-mod} that each quotient \(\mfrac{\mathcal{M}_{i j
+ + 1}}{\mathcal{M}_{i j}}\) is a weight module. Hence
\(\mathcal{M}^{\operatorname{ss}}\) is a weight module. Furtheremore, given
- \(\mu \in \mathfrak{h}^*\)
+ \(\mu \in \lambda_k + Q\)
\[
\mathcal{M}_\mu^{\operatorname{ss}}
= \bigoplus_{i j}
\left(
- \mfrac{\mathcal{M}_{j + 1}[\lambda_i]}{\mathcal{M}_j[\lambda_i]}
+ \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
\right)_\mu
- \cong \bigoplus_{i j}
- \mfrac{\mathcal{M}_{j + 1}[\lambda_i]_\mu}
- {\mathcal{M}_j[\lambda_i]_\mu}
+ = \bigoplus_j
+ \left(
+ \mfrac{\mathcal{M}_{k j + 1}}{\mathcal{M}_{k j}}
+ \right)_\mu
+ \cong \bigoplus_j
+ \mfrac{(\mathcal{M}_{k j + 1})_\mu}
+ {(\mathcal{M}_{k j})_\mu}
\]
In particular,
\[
\dim \mathcal{M}_\mu^{\operatorname{ss}}
- = \sum_{i j}
- \dim \mathcal{M}_{j + 1}[\lambda_i]_\mu
- - \dim \mathcal{M}_j[\lambda_i]_\mu
- = \sum_i \dim \mathcal{M}[\lambda_i]_\mu
+ = \sum_j
+ \dim (\mathcal{M}_{k j + 1})_\mu
+ - \dim (\mathcal{M}_{k j})_\mu
+ = \dim \mathcal{M}[\lambda_k]_\mu
= \dim \mathcal{M}_\mu
= d
\]
@@ -233,13 +236,10 @@
number
\[
\operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu^{\operatorname{ss}}})
- = \sum_{i j}
- \operatorname{Tr}
- (u\!\restriction_{\mathcal{M}_{j + 1}[\lambda_i]_\mu})
- - \operatorname{Tr}
- (u\!\restriction_{\mathcal{M}_j[\lambda_i]_\mu})
- = \sum_{i}
- \operatorname{Tr}(u\!\restriction_{\mathcal{M}[\lambda_i]_\mu})
+ = \sum_j
+ \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j + 1})_\mu})
+ - \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j})_\mu})
+ = \operatorname{Tr}(u\!\restriction_{\mathcal{M}[\lambda_k]_\mu})
= \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
\]
is polynomial in \(\mu \in \mathfrak{h}^*\).