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: c45df8a9192cfe231dec86c4260918c99edd69de
Parent: e644c3d08e4e74de435432c02a72ae0d6c9f7bc8
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 1 Jul 2023 19:12:31 +0000

Added some comments on the proof of a proposition

Diffstat

2 files changed, 17 insertions, 10 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/coherent-families.tex	25	16	9
Modified	sections/fin-dim-simple.tex	2	1	1

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -152,7 +152,7 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
 
 % TODO: Revise the notation for this? I don't really like calling this
 % bijection m
-\begin{proposition}
+\begin{proposition}\label{thm:better-sp(2n)-parameters}
   The map
   \begin{align*}
     m : \mathfrak{h}^* & \to K^n \\
@@ -292,7 +292,7 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
 
 % TODO: Revise the notation for this? I don't really like calling this
 % bijection m
-\begin{proposition}
+\begin{proposition}\label{thm:better-sl(n)-parameters}
   The map
   \begin{align*}
     m : \mathfrak{h}^* &
@@ -314,13 +314,20 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   \emph{not} ordered, but becomes ordered after removing one term.
 \end{proposition}
 
-% TODOO: Note that this prove is similar to the previous one, and that the
-% equivariance of the map follows from the nature of the isomorphism W ≅ S_n
-% TODO: The normalization constant 2n is choosen because
-% λ(H_β) = 2n κ(λ, β) and m(λ) is thus uniquely characterized by the fact that
-% (λ + ρ)(H_β_i) = m(λ)_i - m(λ)_i+1. This comes in when verifying the
-% equivalence between the conditions of the previous lemma and those described
-% in the theorem
+% TODO: Note beforehand that κ(H_α, ⋅) is always a multiple of α. This is
+% perhaps better explained when defining H_α
+The proof of this result is very similar to that of
+Proposition~\ref{thm:better-sp(2n)-parameters} in spirit: the equivariance of
+the map \(m : \mathfrak{h}^* \to \{ \mathfrak{sl}_n\textrm{-sequences} \}\)
+follows from the nature of the isomorphism \(W \cong S_n\), as described in
+Example~\ref{ex:sl-weyl-group}. The number \(2 n\) is a normalization constant
+chosen because \(\lambda(H_\beta) = 2 n \, \kappa(\lambda, \beta)\) for all
+\(\lambda \in \mathfrak{h}^*\) and \(\beta \in \Sigma\). Hence \(m(\lambda)\)
+is uniquely characterized by the property that \((\lambda + \rho)(H_{\beta_i})
+= m(\lambda)_i - m(\lambda)_{i+1}\) for all \(i\), which is relevant to the
+proof of the equivalence between the contiditions of
+Lemma~\ref{thm:sl-bounded-weights} and those explained in the last statement of
+Proposition~\ref{thm:better-sl(n)-parameters}.
 
 % TODO: Explain the significance of 𝓑 + and 𝓑 -: these are the subsets whose
 % union corresponds to condition (i)

diff --git a/sections/fin-dim-simple.tex b/sections/fin-dim-simple.tex
@@ -672,7 +672,7 @@ canonical action of \(W\) on \(\mathfrak{h}\).
 
 This already allow us to compute some examples of Weyl groups.
 
-\begin{example}
+\begin{example}\label{ex:sl-weyl-group}
   Suppose \(\mathfrak{g} = \mathfrak{sl}_n(K)\) and \(\mathfrak{h} \subset
   \mathfrak{g}\) is as in Example~\ref{ex:cartan-of-sl}. Let \(\epsilon_1,
   \ldots, \epsilon_n \in \mathfrak{h}^*\) be as in