lie-algebras-and-their-representations

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 \(\mu \in \mathfrak{h}^*\).