lie-algebras-and-their-representations

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