diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -173,6 +173,19 @@ self-evident: we have just provided a complete description of the action of
left-most eigenvalue of \(h\) is precisely \(n - 2 (m - 1) = -n\).
\end{proof}
+Visually, the situation it thus
+\begin{center}
+ \begin{tikzcd}
+ V_{-n} \arrow[bend left=60]{r}{e}
+ & V_{- n + 2} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}{f}
+ & V_{- n + 4} \arrow[bend left=60]{r} \arrow[bend left=60]{l}{f}
+ & \cdots \arrow[bend left=60]{r} \arrow[bend left=60]{l}
+ & V_{n - 4} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}
+ & V_{n - 2} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}{f}
+ & V_n \arrow[bend left=60]{l}{f}
+ \end{tikzcd}
+\end{center}
+
Corollary~\ref{thm:sl2-find-weights} can be used to find the eigenvalues of the
action of \(h\) on an arbitrary finite-dimensional
\(\mathfrak{sl}_2(K)\)-module. Namely, if \(V\) and \(W\) are representations
@@ -350,7 +363,7 @@ Visually we may draw
\begin{figure}[h]
\centering
- \begin{tikzpicture}[scale=2.5]
+ \begin{tikzpicture}[scale=2]
\begin{rootSystem}{A}
\filldraw[black] \weight{0}{0} circle (.5pt);
\node[black, above right] at \weight{0}{0} {\small$0$};
@@ -396,7 +409,7 @@ For instance \(\mathfrak{sl}_3(K)_{\alpha_1 - \alpha_3}\) will act on the
adjoint representation of \(\mathfrak{sl}_3(K)\) via
\begin{figure}[h]
\centering
- \begin{tikzpicture}[scale=2.5]
+ \begin{tikzpicture}[scale=2]
\begin{rootSystem}{A}
\wt[black]{0}{0}
\wt[black]{-1}{2}
@@ -655,11 +668,9 @@ reproduce these steps in the context of \(\mathfrak{sl}_3(K)\) by fixing a
direction in the plane an considering the weight lying the furthest in that
direction.
-\newpage
-
For instance, let's say we fix the direction
\begin{center}
- \begin{tikzpicture}[scale=2.5]
+ \begin{tikzpicture}[scale=2]
\begin{rootSystem}{A}
\wt[black]{0}{0}
\wt[black]{-1}{2}
@@ -819,8 +830,6 @@ picture is now
\end{tikzpicture}
\end{center}
-\newpage
-
Needless to say, we could keep applying this method to the weights at the ends
of our string, arriving at
\begin{center}
@@ -1106,7 +1115,7 @@ representation turns out to be quite simple.
\(\alpha_1\), \(\alpha_2\) and \(\alpha_3\) respectively. Hence the weight
diagram of \(K^3\) is
\begin{center}
- \begin{tikzpicture}[scale=2.5]
+ \begin{tikzpicture}[scale=2]
\AutoSizeWeightLatticefalse
\begin{rootSystem}{A}
\weightLattice{2}
@@ -1126,7 +1135,7 @@ representation turns out to be quite simple.
\mathfrak{h}\), so that the weights of \((K^3)^*\) are precisely the
opposites of the weights of \(K^3\). In other words,
\begin{center}
- \begin{tikzpicture}[scale=2.5]
+ \begin{tikzpicture}[scale=2]
\AutoSizeWeightLatticefalse
\begin{rootSystem}{A}
\weightLattice{2}