lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: 5366d0dd99ebc8c2b7e80ca9caf160443bfd0d1d
Parent: 967dec4c13dc1048b4aa62769f218be5ca41d88a
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 10 Nov 2022 13:40:23 +0000

Added a footnote

Diffstat

1 file changed, 4 insertions, 1 deletion

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	5	4	1

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -509,7 +509,10 @@
   The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
   from corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
   corollary that \strong{(iii)} implies \strong{(ii)}. All it's left is to show
-  \strong{(ii)} implies \strong{(iii)}.
+  \strong{(ii)} implies \strong{(iii)}\footnote{This isn't already clear from
+  corollary~\ref{thm:cuspidal-mod-equivs} because, at first glance,
+  $\mathcal{M}[\lambda]$ may not be irreducible for some $\lambda$ satisfying
+  \strong{(ii)}. We will show this is never the case.}.
 
   Suppose \(F_\alpha\) acts injectively in the subrepresentation
   \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since