diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -509,14 +509,14 @@ example~\ref{ex:gln-inclusions} -- restricts to an injective homomorphism
Our first observation is that, since the root spaces act by translation, the
subspace
\[
- \bigoplus_{k \in \mathbb{Z}} V_{\lambda + k (\alpha_1 - \alpha_2)},
+ \bigoplus_{k \in \mathbb{Z}} V_{\lambda - k (\alpha_1 - \alpha_2)},
\]
must be invariant under the action of \(E_{1 2}\) and \(E_{2 1}\) for all
-\(\lambda \in \mathfrak{h}^*\). This goes to show \(\bigoplus_k V_{\lambda + k
+\(\lambda \in \mathfrak{h}^*\). This goes to show \(\bigoplus_k V_{\lambda - k
(\alpha_1 - \alpha_2)}\) is a \(\mathfrak{sl}_2(K)\)-submodule of \(V\) for all
weights \(\lambda\) of \(V\). Furtheremore, one can easily see that the
eigenspace of the eigenspace \(\lambda(H) - 2k\) of \(h\) in \(W\) is precisely
-the weight space \(V_{\lambda + k (\alpha_2 - \alpha_1)}\).
+the weight space \(V_{\lambda - k (\alpha_2 - \alpha_1)}\).
Visually,
\begin{center}
@@ -549,15 +549,16 @@ Visually,
In general, we find\dots
\begin{proposition}
- The subalgebra \(\mathfrak{s}_{\alpha_i - \alpha_j} = K \langle E_{i j}, E_{j
- i}, [E_{i j}, E_{j i}] \rangle\) is isomorphic to \(\mathfrak{sl}_2(K)\).
- In addition, given weight \(\lambda \in \mathfrak{h}^*\) of \(V\), the space
+ Given \(i < j\), the subalgebra \(\mathfrak{s}_{\alpha_i - \alpha_j} = K
+ \langle E_{i j}, E_{j i}, [E_{i j}, E_{j i}] \rangle\) is isomorphic to
+ \(\mathfrak{sl}_2(K)\). In addition, given weight \(\lambda \in
+ \mathfrak{h}^*\) of \(V\), the space
\[
- W = \bigoplus_{k \in \mathbb{Z}} V_{\lambda + k (\alpha_i - \alpha_j)}
+ W = \bigoplus_{k \in \mathbb{Z}} V_{\lambda - k (\alpha_i - \alpha_j)}
\]
is invariant under the action of \(\mathfrak{s}_{i j}\) and
\[
- V_{\lambda + k (\alpha_i - \alpha_j)}
+ V_{\lambda - k (\alpha_i - \alpha_j)}
= W_{\lambda([E_{i j}, E_{j i}]) - 2k}
\]
\end{proposition}
@@ -574,21 +575,21 @@ In general, we find\dots
To see that \(W\) is invariant under the action of \(\mathfrak{s}_{i j}\), it
suffices to notice \(E_{i j}\) and \(E_{j i}\) map \(v \in V_{\lambda + k
- (\alpha_i - \alpha_j)}\) to \(E_{i j} v \in V_{\lambda + (k + 1) (\alpha_i -
- \alpha_j)}\) and \(E_{j i} v \in V_{\lambda + (k - 1) (\alpha_i -
+ (\alpha_i - \alpha_j)}\) to \(E_{i j} v \in V_{\lambda - (k - 1) (\alpha_i -
+ \alpha_j)}\) and \(E_{j i} v \in V_{\lambda - (k + 1) (\alpha_i -
\alpha_j)}\). Moreover,
\[
- (\lambda + k (\alpha_i - \alpha_j))(H)
- = \lambda([E_{i j}, E_{j i}]) + k (-1 - 1)
+ (\lambda - k (\alpha_i - \alpha_j))([E_{i j}, E_{j i}])
+ = \lambda([E_{i j}, E_{j i}]) - k (1 - (-1))
= \lambda([E_{i j}, E_{j i}]) - 2 k,
\]
- which goes to show \(V_{\lambda + k (\alpha_i - \alpha_j)} \subset
+ which goes to show \(V_{\lambda - k (\alpha_i - \alpha_j)} \subset
W_{\lambda([E_{i j}, E_{j i}]) - 2k}\). On the other hand, if we suppose \(0
- < \dim V_{\lambda + k (\alpha_i - \alpha_j)} < \dim W_{\lambda([E_{i j}, E_{j
+ < \dim V_{\lambda - k (\alpha_i - \alpha_j)} < \dim W_{\lambda([E_{i j}, E_{j
i}]) - 2 k}\) for some \(k\) we arrive at
\[
\dim W
- = \sum_k \dim V_{\lambda + k (\alpha_i - \alpha_j)}
+ = \sum_k \dim V_{\lambda - k (\alpha_i - \alpha_j)}
< \sum_k \dim W_{\lambda([E_{i j}, E_{j i}]) - 2k}
= \dim W,
\]