lie-algebras-and-their-representations

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -972,13 +972,23 @@ This final picture is known as \emph{the weight diagram of \(M\)}. Finally\dots
 
 Having found all of the weights of \(M\), the only thing we are missing is an
 existence and uniqueness theorem analogous to
-Theorem~\ref{thm:sl2-exist-unique}. In other words, our next goal is
-establishing\dots
+Theorem~\ref{thm:sl2-exist-unique}. It is clear from the symmetries of the
+locus of weights found in Theorem~\ref{thm:sl3-irr-weights-class} that if
+\(\lambda \in P\) is the highest weight of some finite-dimensional simple
+\(\mathfrak{sl}_3(K)\)-module \(M\) then \(\lambda\) lies in the cone
+\(\mathbb{N} \langle \alpha_1, - \alpha_3 \rangle\). What's perhaps more
+surprising is the fact that this condition is sufficient for the existance of
+such a \(M\). In other words, our next goal is establishing\dots
+
+\begin{definition}
+  An element \(\lambda \in P\) is called \emph{dominant} if it lies in the cone
+  \(\mathbb{N} \langle \alpha_1, - \alpha_3 \rangle\).
+\end{definition}
 
 \begin{theorem}\label{thm:sl3-existence-uniqueness}
-  For each pair of non-negative integers \(k\) and \(\ell\), there exists
-  precisely one finite-dimensional simple \(\mathfrak{sl}_3(K)\)-module \(M\)
-  whose highest weight is \(k \alpha_1 - \ell \alpha_3\).
+  For each dominant \(\lambda \in P\), there exists precisely one
+  finite-dimensional simple \(\mathfrak{sl}_3(K)\)-module \(M\) whose highest
+  weight is \(\lambda\).
 \end{theorem}
 
 To proceed further we once again refer to the approach we employed in the case
@@ -1028,7 +1038,7 @@ Specifically\dots
   We begin by analyzing the case of \(E_{1 2}\). We have
   \[
     \begin{split}
-      E_{1 2} \cdot (E_{2 1}^a E_{3 1}^b E_{3 2}^c \cdot m)
+      E_{1 2} E_{2 1}^a E_{3 1}^b E_{3 2}^c \cdot m
       & = ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2})
           E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c \cdot m \\
       & = E_{2 1} ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2})
@@ -1113,15 +1123,17 @@ The same argument also goes to show\dots
 \end{corollary}
 
 This is very interesting to us since it implies that finding \emph{any}
-finite-dimensional module whose highest weight is \(k \alpha_1 - \ell
-\alpha_3\) is enough for establishing the ``existence'' part of
+finite-dimensional module whose highest weight is \(\lambda\) is enough for
+establishing the ``existence'' part of
 Theorem~\ref{thm:sl3-existence-uniqueness}. Moreover, constructing such a
 module turns out to be quite simple.
 
 \begin{proof}[Proof of existence]
-  Consider the natural \(\mathfrak{sl}_3(K)\)-module \(K^3\). We claim that the
-  highest weight of \(\operatorname{Sym}^k K^3 \otimes \operatorname{Sym}^\ell
-  (K^3)^*\) is \(k \alpha_1 - \ell \alpha_3\).
+  Take \(\lambda = k \alpha_1 - \ell \alpha_3 \in P\) with \(k, \ell \ge 0\),
+  so that \(\lambda\) is dominant. Consider the natural
+  \(\mathfrak{sl}_3(K)\)-module \(K^3\). We claim that the highest weight of
+  \(\operatorname{Sym}^k K^3 \otimes \operatorname{Sym}^\ell (K^3)^*\) is
+  \(\lambda\).
 
   First of all, notice that the weight vector of \(K^3\) are the canonical
   basis elements \(e_1\), \(e_2\) and \(e_3\), whose corresponding weights are
@@ -1164,8 +1176,8 @@ module turns out to be quite simple.
   is the weight diagram of \((K^3)^*\) and \(\alpha_3\) is the highest weight
   of \((K^3)^*\).
 
-  On the other hand if we fix two \(\mathfrak{sl}_3(K)\)-modules \(N\)
-  and \(L\), by computing
+  On the other hand if we fix two \(\mathfrak{sl}_3(K)\)-modules \(N\) and
+  \(L\), by computing
   \[
     \begin{split}
       H \cdot (n \otimes l)
@@ -1183,7 +1195,7 @@ module turns out to be quite simple.
   \alpha_3\) respectively -- with highest weight vectors \(e_1^k\) and
   \(f_3^\ell\). Furthermore, by the same token the highest weight of
   \(\operatorname{Sym}^k K^3 \otimes \operatorname{Sym}^\ell (K^3)^*\) must be
-  \(k e_1 - \ell e_3\) -- with highest weight vector \(e_1^k \otimes
+  \(\lambda = k e_1 - \ell e_3\) -- with highest weight vector \(e_1^k \otimes
   f_3^\ell\).
 \end{proof}