lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -319,13 +319,14 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
 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
+follows from the nature of the isomorphism \(W \cong S_n\) as described in
+Example~\ref{ex:sl-weyl-group}, while the rest of the proof amounts to simple
+technical verifications. 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}.