lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -652,12 +652,12 @@ the argument used for \(\mathfrak{sl}_3(K)\). Namely\dots
 \end{proposition}
 
 The proof of proposition~\ref{thm:irr-subrep-generated-by-vec} is very similar
-to that of proposition~\ref{thm:sl3-positive-roots-span-all-irr-rep} in
-spirit: we use the commutator relations of \(\mathfrak{g}\) to inductively show
-that the subspace spanned by the images of a highest weight vector under
-successive applications of negative root vectors is invariant under the action
-of \(\mathfrak{g}\) -- please refer to \cite{fulton-harris} for further
-details. Of course, what we are really interested in is\dots
+in spirit to that of proposition~\ref{thm:sl3-positive-roots-span-all-irr-rep}:
+we use the commutator relations of \(\mathfrak{g}\) to inductively show that
+the subspace spanned by the images of a highest weight vector under successive
+applications of negative root vectors is invariant under the action of
+\(\mathfrak{g}\) -- please refer to \cite{fulton-harris} for further details.
+Of course, what we are really interested in is\dots
 
 \begin{corollary}
   Let \(V\) and \(W\) be finite-dimensional irreducible
@@ -864,10 +864,10 @@ whose highest weight is \(\lambda\).
     V = \bigoplus_\mu V_\mu = \bigoplus_\mu M(\lambda)_\mu \cap V
   \]
 
-  Since \(M(\lambda) = \mathcal{U}(\mathfrak{g}) \cdot v^+\), \(V\) is a proper
-  subrepresentation then \(v^+ \notin V\). Hence any proper submodule lies in
-  the sum of weight spaces other than \(M(\lambda)_\lambda\), so the sum
-  \(N(\lambda)\) of all such submodules is still proper. This implies
+  Since \(M(\lambda) = \mathcal{U}(\mathfrak{g}) \cdot v^+\), if \(V\) is a
+  proper subrepresentation then \(v^+ \notin V\). Hence any proper submodule
+  lies in the sum of weight spaces other than \(M(\lambda)_\lambda\), so the
+  sum \(N(\lambda)\) of all such submodules is still proper. This implies
   \(N(\lambda)\) is the unique maximal subrepresentation of \(M(\lambda)\) and
   \(L(\lambda) = \sfrac{M(\lambda)}{N(\lambda)}\) is its unique irreducible
   quotient.
@@ -916,7 +916,7 @@ are really interested in is\dots
 \begin{proof}
   Let \(V = L(\lambda)\). It suffices to show that its highest weight is
   \(\lambda\). We have already seen that \(v^+ \in M(\lambda)_\lambda\) is a
-  highest weight vector. Now since \(v\) lies outside of the maximal
+  highest weight vector. Now since \(v^+\) lies outside of the maximal
   subrepresentation of \(M(\lambda)\), the projection \(v^+ + N(\lambda) \in
   V\) is nonzero.