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}