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: df1d4f6972e0041bfb767c73264f052d06ece0a5
Parent: 08ae3830561549c71f98cebb4732a52f68aae30f
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 21 Jun 2023 19:14:12 +0000

Revised the comments on regular and integral sl(n)-sequences

Diffstat

1 file changed, 10 insertions, 11 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/coherent-families.tex	21	10	11

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -256,17 +256,16 @@ It should then be obvious that\dots
   m_n \ \text{is \emph{not} a positive integer}\}\).
 \end{definition}
 
-\begin{definition}
-  A \(\mathfrak{sl}_n\)-sequence \(m\) is called \emph{integral} if \(m_i - m_j
-  \in \mathbb{Z}\) for all \(i\) and \(j\).
-\end{definition}
-
-% TODOO: Discuss the notion of a regular weight beforehand
-\begin{definition}
-  A \(\mathfrak{sl}_n\)-sequence \(m\) is called \emph{regular} if \(m_i \ne
-  m_j\) for all \(i \ne j\). A sequence is called \emph{singular} if it is not
-  regular.
-\end{definition}
+A weight \(\lambda \in \mathfrak{h}^*\) is called \emph{regular} if \((\lambda
++ \rho)(H_\alpha) \ne 0\) for all \(\alpha \in \Delta\). In terms of
+\(\mathfrak{sl}_n\)-sequences, \(\lambda\) is regular if, and only if
+\(m(\lambda)_i \ne m(\lambda)_j\) for all \(i \ne j\). It thus makes sence to
+call a \(\mathfrak{sl}_n\)-sequence regular or singular if \(m_i \ne m_j\) for
+all \(i \ne j\) or \(m_i = m_j\) for some \(i \ne j\), respectively. Similarly,
+\(\lambda\) is integral if, and only if \(m(\lambda)_i - m(\lambda)_j \in
+\mathbb{Z}\) for all \(i\) and \(j\), so it makes sence to call a
+\(\mathfrak{sl}_n\)-sequence \(m\) integral if \(m_i - m_j \in \mathbb{Z}\) for
+all \(i\) and \(j\).
 
 % TODO: Add notes on what are the sets W ⋅m ∩ 𝓟  : the connected component of
 % a given element is contained in its orbit, but a given orbit may contain