lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -291,15 +291,15 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   \begin{align*}
     \mathfrak{h}^* & \to K^n \\
     \lambda &
-    \mapsto (\kappa(\epsilon_1, \lambda), \ldots, \kappa(\epsilon_1, \lambda))
+    \mapsto (\kappa(\epsilon_1, \lambda), \ldots, \kappa(\epsilon_n, \lambda))
   \end{align*}
   is equivariant with respect to the natural action of \(W\) on
   \(\mathfrak{h}^*\). But this also clear from the isomorphism \(W \cong S_n
   \ltimes (\mathbb{Z}/2\mathbb{Z})^n\), as described in
   Example~\ref{ex:sp-weyl-group}: \((\sigma_i, (\bar 0, \ldots, \bar 0)) =
   \sigma_{\beta_i}\) permutes \(\epsilon_i\) and \(\epsilon_{i + 1}\) for \(i <
-  n\) and \((1, (\bar 0, \ldots, \bar 0, \bar 1) = \sigma_{\beta_n}\) flips the
-  sign of \(\epsilon_n\). Hence \(m(\sigma_{\beta_i} \cdot \epsilon_j) =
+  n\) and \((1, (\bar 0, \ldots, \bar 0, \bar 1)) = \sigma_{\beta_n}\) flips
+  the sign of \(\epsilon_n\). Hence \(m(\sigma_{\beta_i} \cdot \epsilon_j) =
   \sigma_{\beta_i} \cdot m(\epsilon_j)\) for all \(i\) and \(j\). Since \(W\)
   is generated by the \(\sigma_{\beta_i}\), this implies that the required map
   is equivariant.