diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -364,10 +364,10 @@ the eigenvalues of the action of \(h\) on a simple
In the case of \(\mathfrak{sl}_3(K)\), a simple calculation shows that if \([H,
X]\) is scalar multiple of \(X\) for all \(H \in \mathfrak{h}\) then all but
one entry of \(X\) are zero. Hence the eigenvectors of the adjoint action of
-\(\mathfrak{h}\) are \(E_{i j}\) and its eigenvalues are \(\alpha_i -
-\alpha_j\), where
+\(\mathfrak{h}\) are \(E_{i j}\) and its eigenvalues are \(\epsilon_i -
+\epsilon_j\), where
\[
- \alpha_i
+ \epsilon_i
\begin{pmatrix}
a_1 & 0 & 0 \\
0 & a_2 & 0 \\
@@ -390,15 +390,15 @@ Visually we may draw
\wt[black]{-1}{-1}
\wt[black]{2}{-1}
\wt[black]{1}{-2}
- \node[above] at \weight{-1}{2} {$\alpha_2 - \alpha_3$};
- \node[left] at \weight{-2}{1} {$\alpha_2 - \alpha_1$};
- \node[right] at \weight{1}{1} {$\alpha_1 - \alpha_3$};
- \node[left] at \weight{-1}{-1} {$\alpha_3 - \alpha_1$};
- \node[right] at \weight{2}{-1} {$\alpha_1 - \alpha_2$};
- \node[below] at \weight{1}{-2} {$\alpha_3 - \alpha_1$};
- \node[black, above] at \weight{1}{0} {$\alpha_1$};
- \node[black, above] at \weight{-1}{1} {$\alpha_2$};
- \node[black, above] at \weight{0}{-1} {$\alpha_3$};
+ \node[above] at \weight{-1}{2} {$\epsilon_2 - \epsilon_3$};
+ \node[left] at \weight{-2}{1} {$\epsilon_2 - \epsilon_1$};
+ \node[right] at \weight{1}{1} {$\epsilon_1 - \epsilon_3$};
+ \node[left] at \weight{-1}{-1} {$\epsilon_3 - \epsilon_1$};
+ \node[right] at \weight{2}{-1} {$\epsilon_1 - \epsilon_2$};
+ \node[below] at \weight{1}{-2} {$\epsilon_3 - \epsilon_1$};
+ \node[black, above] at \weight{1}{0} {$\epsilon_1$};
+ \node[black, above] at \weight{-1}{1} {$\epsilon_2$};
+ \node[black, above] at \weight{0}{-1} {$\epsilon_3$};
\filldraw[black] \weight{1}{0} circle (.5pt);
\filldraw[black] \weight{-1}{1} circle (.5pt);
\filldraw[black] \weight{0}{-1} circle (.5pt);
@@ -422,7 +422,7 @@ so that \(X\) carries \(m\) to \(M_{\lambda + \alpha}\). In other words,
\(\mathfrak{sl}_3(k)_\alpha\) \emph{acts on \(M\) by translating vectors
between eigenspaces}.
-For instance \(\mathfrak{sl}_3(K)_{\alpha_1 - \alpha_3}\) will act on the
+For instance \(\mathfrak{sl}_3(K)_{\epsilon_1 - \epsilon_3}\) will act on the
adjoint \(\mathfrak{sl}_3(K)\)-modules via
\begin{figure}[h]
\centering
@@ -449,7 +449,7 @@ This is again entirely analogous to the situation we observed in
\begin{theorem}\label{thm:sl3-weights-congruent-mod-root}
The eigenvalues of the action of \(\mathfrak{h}\) on a simple
\(\mathfrak{sl}_3(K)\)-module \(M\) differ from one another by integral
- linear combinations of the eigenvalues \(\alpha_i - \alpha_j\) of the adjoint
+ linear combinations of the eigenvalues \(\epsilon_i - \epsilon_j\) of the adjoint
action of \(\mathfrak{h}\) on \(\mathfrak{sl}_3(K)\).
\end{theorem}
@@ -458,7 +458,7 @@ This is again entirely analogous to the situation we observed in
\(\mathfrak{sl}_2(K)\): it suffices to note that if we fix some eigenvalue
\(\lambda\) of \(\mathfrak{h}\) and let \(i\) and \(j\) vary then
\[
- \bigoplus_{i j} M_{\lambda + \alpha_i - \alpha_j}
+ \bigoplus_{i j} M_{\lambda + \epsilon_i - \epsilon_j}
\]
is an invariant subspace of \(M\).
\end{proof}
@@ -487,7 +487,7 @@ It is clear from our previous discussion that the weights of the adjoint
Theorem~\ref{thm:sl3-weights-congruent-mod-root} can thus be restated as\dots
\begin{definition}\index{weights!root lattice}
- The lattice \(Q = \mathbb{Z} \langle \alpha_i - \alpha_j : i, j = 1, 2, 3
+ The lattice \(Q = \mathbb{Z} \langle \epsilon_i - \epsilon_j : i, j = 1, 2, 3
\rangle\) is called \emph{the root lattice of \(\mathfrak{sl}_3(K)\)}.
\end{definition}
@@ -513,15 +513,15 @@ Example~\ref{ex:gln-inclusions} -- restricts to an injective homomorphism
Our first observation is that, since the root spaces act by translation, the
subspace
\[
- \bigoplus_{k \in \mathbb{Z}} M_{\lambda - k (\alpha_1 - \alpha_2)},
+ \bigoplus_{k \in \mathbb{Z}} M_{\lambda - k (\epsilon_1 - \epsilon_2)},
\]
must be invariant under the action of \(E_{1 2}\) and \(E_{2 1}\) for all
\(\lambda \in \mathfrak{h}^*\). This goes to show \(\bigoplus_k M_{\lambda - k
-(\alpha_1 - \alpha_2)}\) is a \(\mathfrak{sl}_2(K)\)-submodule of \(M\) for all
+(\epsilon_1 - \epsilon_2)}\) is a \(\mathfrak{sl}_2(K)\)-submodule of \(M\) for all
weights \(\lambda\) of \(M\). Furthermore, one can easily see that the
eigenspace of the action of \(h\) on \(\bigoplus_{k \in \mathbb{Z}} M_{\lambda
-- k (\alpha_1 - \alpha_2)}\) associated with the eigenvalue \(\lambda(H) - 2k\)
-is precisely the weight space \(M_{\lambda - k (\alpha_2 - \alpha_1)}\).
+- k (\epsilon_1 - \epsilon_2)}\) associated with the eigenvalue \(\lambda(H) - 2k\)
+is precisely the weight space \(M_{\lambda - k (\epsilon_2 - \epsilon_1)}\).
Visually,
\begin{center}
@@ -559,11 +559,11 @@ In general, we find\dots
\(\mathfrak{sl}_2(K)\). In addition, given a weight \(\lambda \in
\mathfrak{h}^*\) of \(M\), the space
\[
- N = \bigoplus_{k \in \mathbb{Z}} M_{\lambda - k (\alpha_i - \alpha_j)}
+ N = \bigoplus_{k \in \mathbb{Z}} M_{\lambda - k (\epsilon_i - \epsilon_j)}
\]
is invariant under the action of \(\mathfrak{s}_{i j}\) and
\[
- M_{\lambda - k (\alpha_i - \alpha_j)}
+ M_{\lambda - k (\epsilon_i - \epsilon_j)}
= N_{\lambda([E_{i j}, E_{j i}]) - 2k}
\]
\end{proposition}
@@ -604,21 +604,21 @@ In general, we find\dots
To see that \(N\) is invariant under the action of \(\mathfrak{s}_{i j}\), it
suffices to notice \(E_{i j}\) and \(E_{j i}\) map \(m \in M_{\lambda - k
- (\alpha_i - \alpha_j)}\) to \(E_{i j} \cdot m \in M_{\lambda - (k - 1) (\alpha_i -
- \alpha_j)}\) and \(E_{j i} \cdot m \in M_{\lambda - (k + 1) (\alpha_i -
- \alpha_j)}\), respectively. Moreover,
+ (\epsilon_i - \epsilon_j)}\) to \(E_{i j} \cdot m \in M_{\lambda - (k - 1) (\epsilon_i -
+ \epsilon_j)}\) and \(E_{j i} \cdot m \in M_{\lambda - (k + 1) (\epsilon_i -
+ \epsilon_j)}\), respectively. Moreover,
\[
- (\lambda - k (\alpha_i - \alpha_j))([E_{i j}, E_{j i}])
+ (\lambda - k (\epsilon_i - \epsilon_j))([E_{i j}, E_{j i}])
= \lambda([E_{i j}, E_{j i}]) - k (1 - (-1))
= \lambda([E_{i j}, E_{j i}]) - 2 k,
\]
- which goes to show \(M_{\lambda - k (\alpha_i - \alpha_j)} \subset
+ which goes to show \(M_{\lambda - k (\epsilon_i - \epsilon_j)} \subset
N_{\lambda([E_{i j}, E_{j i}]) - 2k}\). On the other hand, if we suppose \(0
- < \dim M_{\lambda - k (\alpha_i - \alpha_j)} < \dim N_{\lambda([E_{i j}, E_{j
+ < \dim M_{\lambda - k (\epsilon_i - \epsilon_j)} < \dim N_{\lambda([E_{i j}, E_{j
i}]) - 2 k}\) for some \(k\) we arrive at
\[
\dim N
- = \sum_k \dim M_{\lambda - k (\alpha_i - \alpha_j)}
+ = \sum_k \dim M_{\lambda - k (\epsilon_i - \epsilon_j)}
< \sum_k \dim N_{\lambda([E_{i j}, E_{j i}]) - 2k}
= \dim N,
\]
@@ -628,7 +628,7 @@ In general, we find\dots
As a first consequence of this, we show\dots
\begin{definition}\index{weights!weight lattice}
- The lattice \(P = \mathbb{Z} \langle \alpha_1, \alpha_2, \alpha_3 \rangle\)
+ The lattice \(P = \mathbb{Z} \langle \epsilon_1, \epsilon_2, \epsilon_3 \rangle\)
is called \emph{the weight lattice of \(\mathfrak{sl}_3(K)\)}.
\end{definition}
@@ -664,8 +664,8 @@ As a first consequence of this, we show\dots
=
a \lambda([E_{1 3}, E_{3 1}]) + b \lambda([E_{2 3}, E_{3 2}]),
\]
- which is to say \(\lambda = \lambda([E_{1 3}, E_{3 1}]) \alpha_1 +
- \lambda([E_{2 3}, E_{3 2}]) \alpha_2 \in P\).
+ which is to say \(\lambda = \lambda([E_{1 3}, E_{3 1}]) \epsilon_1 +
+ \lambda([E_{2 3}, E_{3 2}]) \epsilon_2 \in P\).
\end{proof}
There is a clear parallel between the case of \(\mathfrak{sl}_3(K)\) and that
@@ -675,17 +675,17 @@ the sublattice \(Q = 2 \mathbb{Z}\).
Among other things, this last result goes to show that the diagrams we have
been drawing are in fact consistent with the theory we have developed. Namely,
-since all weights lie in the rational span of \(\{\alpha_1, \alpha_2,
-\alpha_3\}\), we may as well draw them in the Cartesian plane. In fact, the
+since all weights lie in the rational span of \(\{\epsilon_1, \epsilon_2,
+\epsilon_3\}\), we may as well draw them in the Cartesian plane. In fact, the
attentive reader may notice that \(\kappa(E_{1 2}, E_{2 3}) = - \sfrac{1}{2}\),
so that the angle -- with respect to the Killing form \(\kappa\) -- between the
root vectors \(E_{1 2}\) and \(E_{2 3}\) is precisely the same as the angle
-between the points representing their roots \(\alpha_1 - \alpha_2\) and
-\(\alpha_2 - \alpha_3\) in the Cartesian plane. Since \(\alpha_1 - \alpha_2\)
-and \(\alpha_2 - \alpha_3\) span \(\mathfrak{h}^*\), this implies the diagrams
+between the points representing their roots \(\epsilon_1 - \epsilon_2\) and
+\(\epsilon_2 - \epsilon_3\) in the Cartesian plane. Since \(\epsilon_1 - \epsilon_2\)
+and \(\epsilon_2 - \epsilon_3\) span \(\mathfrak{h}^*\), this implies the diagrams
we've been drawing are given by an isometry \(\mathbb{Q} P \isoto
\mathbb{Q}^2\), where \(\mathbb{Q} P\) is endowed with the bilinear form
-defined by \((\alpha_i - \alpha_j, \alpha_k - \alpha_\ell) \mapsto \kappa(E_{i
+defined by \((\epsilon_i - \epsilon_j, \epsilon_k - \alpha_\ell) \mapsto \kappa(E_{i
j}, E_{k \ell})\) -- which we denote by \(\kappa\) as well.
To proceed we once more refer to the previously established framework: next we
@@ -751,8 +751,8 @@ weight placed the furthest in the direction we chose. Given our previous
assertion that the root spaces of \(\mathfrak{sl}_3(K)\) act on the weight
spaces of \(M\) via translation, this implies that \(E_{1 2}\), \(E_{1 3}\) and
\(E_{2 3}\) all annihilate \(M_\lambda\), or otherwise one of \(M_{\lambda +
-\alpha_1 - \alpha_2}\), \(M_{\lambda + \alpha_1 - \alpha_3}\) and \(M_{\lambda
-+ \alpha_2 - \alpha_3}\) would be nonzero -- which contradicts the hypothesis
+\epsilon_1 - \epsilon_2}\), \(M_{\lambda + \epsilon_1 - \epsilon_3}\) and \(M_{\lambda
++ \epsilon_2 - \epsilon_3}\) would be nonzero -- which contradicts the hypothesis
that \(\lambda\) lies the furthest in the direction we chose. In other
words\dots
@@ -763,8 +763,8 @@ words\dots
\begin{proof}
It suffices to note that the positive roots of \(\mathfrak{sl}_3(K)\) are
- precisely \(\alpha_1 - \alpha_2\), \(\alpha_1 - \alpha_3\) and \(\alpha_2 -
- \alpha_3\), with root vectors \(E_{1 2}\), \(E_{1 3}\) and \(E_{2 3}\),
+ precisely \(\epsilon_1 - \epsilon_2\), \(\epsilon_1 - \epsilon_3\) and \(\epsilon_2 -
+ \epsilon_3\), with root vectors \(E_{1 2}\), \(E_{1 3}\) and \(E_{2 3}\),
respectively.
\end{proof}
@@ -805,14 +805,14 @@ from \(M_\lambda\) by successively applying \(E_{2 1}\).
Notice that \(\lambda([E_{1 2}, E_{2 1}]) \in \mathbb{Z}\) is the right-most
eigenvalue of the \(\mathfrak{sl}_2(K)\)-module \(\bigoplus_{k \in \mathbb{Z}}
-M_{\lambda - k (\alpha_1 - \alpha_2)}\). In particular, \(\lambda([E_{1 2},
+M_{\lambda - k (\epsilon_1 - \epsilon_2)}\). In particular, \(\lambda([E_{1 2},
E_{2 1}])\) must be positive. In addition, since the eigenspace of the
eigenvalue \(\lambda([E_{1 2}, E_{2 1}]) - 2k\) of the action of \(h\) on
-\(\bigoplus_{k \in \mathbb{N}} M_{\lambda - k (\alpha_1 - \alpha_2)}\) is
-\(M_{\lambda - k (\alpha_1 - \alpha_2)}\), the weights of \(M\) appearing the
-string \(\lambda, \lambda + (\alpha_1 - \alpha_2), \ldots, \lambda + k
-(\alpha_1 - \alpha_2), \ldots\) must be symmetric with respect to the line
-\(\kappa(\alpha_1 - \alpha_2, \alpha) = 0\). The picture is thus
+\(\bigoplus_{k \in \mathbb{N}} M_{\lambda - k (\epsilon_1 - \epsilon_2)}\) is
+\(M_{\lambda - k (\epsilon_1 - \epsilon_2)}\), the weights of \(M\) appearing the
+string \(\lambda, \lambda + (\epsilon_1 - \epsilon_2), \ldots, \lambda + k
+(\epsilon_1 - \epsilon_2), \ldots\) must be symmetric with respect to the line
+\(\kappa(\epsilon_1 - \epsilon_2, \alpha) = 0\). The picture is thus
\begin{center}
\begin{tikzpicture}
\AutoSizeWeightLatticefalse
@@ -825,14 +825,14 @@ string \(\lambda, \lambda + (\alpha_1 - \alpha_2), \ldots, \lambda + k
\foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}}
\node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)};
\draw[very thick] \weight{0}{-4} -- \weight{0}{4}
- node[above]{\small\(\kappa(\alpha_1 - \alpha_2, \alpha) = 0\)};
+ node[above]{\small\(\kappa(\epsilon_1 - \epsilon_2, \alpha) = 0\)};
\end{rootSystem}
\end{tikzpicture}
\end{center}
We could apply this same argument to the subspace \(\bigoplus_k M_{\lambda - k
-(\alpha_2 - \alpha_3)}\), so that the weights in this subspace must be
-symmetric with respect to the line \(\kappa(\alpha_2 - \alpha_3, \alpha) = 0\).
+(\epsilon_2 - \epsilon_3)}\), so that the weights in this subspace must be
+symmetric with respect to the line \(\kappa(\epsilon_2 - \epsilon_3, \alpha) = 0\).
The picture is now
\begin{center}
\begin{tikzpicture}
@@ -848,9 +848,9 @@ The picture is now
\foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}}
\node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)};
\draw[very thick] \weight{0}{-4} -- \weight{0}{4}
- node[above]{\small\(\kappa(\alpha_1 - \alpha_2, \alpha) = 0\)};
+ node[above]{\small\(\kappa(\epsilon_1 - \epsilon_2, \alpha) = 0\)};
\draw[very thick] \weight{-4}{0} -- \weight{4}{0}
- node[right]{\small\(\kappa(\alpha_2 - \alpha_3, \alpha) = 0\)};
+ node[right]{\small\(\kappa(\epsilon_2 - \epsilon_3, \alpha) = 0\)};
\end{rootSystem}
\end{tikzpicture}
\end{center}
@@ -973,7 +973,7 @@ This final picture is known as \emph{the weight diagram of \(M\)}. Finally\dots
The weights of \(M\) are precisely the elements of the weight lattice \(P\)
congruent to \(\lambda\) module the sublattice \(Q\) and lying inside hexagon
with vertices the images of \(\lambda\) under the group generated by
- reflections across the lines \(\kappa(\alpha_i - \alpha_j, \alpha) = 0\).
+ reflections across the lines \(\kappa(\epsilon_i - \epsilon_j, \alpha) = 0\).
\end{theorem}
Having found all of the weights of \(M\), the only thing we are missing is an
@@ -982,13 +982,13 @@ 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
+\(\mathbb{N} \langle \epsilon_1, - \epsilon_3 \rangle\). What's perhaps more
surprising is the fact that this condition is sufficient for the existence of
such a \(M\). In other words, our next goal is establishing\dots
\begin{definition}\index{weights!dominant weight}
An element \(\lambda \in P\) is called \emph{dominant} if it lies in the cone
- \(\mathbb{N} \langle \alpha_1, - \alpha_3 \rangle\).
+ \(\mathbb{N} \langle \epsilon_1, - \epsilon_3 \rangle\).
\end{definition}
\begin{theorem}\label{thm:sl3-existence-uniqueness}
@@ -1024,15 +1024,15 @@ Specifically\dots
\(E_{2 1}^a E_{3 1}^b E_{3 1}^c\) for some \(a\), \(b\) and \(c\), so that
\(N\) is spanned by the elements \(E_{2 1}^a E_{3 1}^b E_{3 1}^c \cdot m\).
- Recall that \(E_{i j}\) maps \(M_\mu\) to \(M_{\mu + \alpha_i - \alpha_j}\).
+ Recall that \(E_{i j}\) maps \(M_\mu\) to \(M_{\mu + \epsilon_i - \epsilon_j}\).
In particular, \(E_{2 1}^a E_{3 1}^b E_{3 1}^c \cdot m \in M_{\lambda - a
- (\alpha_1 - \alpha_2) - b (\alpha_1 - \alpha_3) - c (\alpha_2 - \alpha_3)}\).
+ (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3)}\).
In other words,
\[
H E_{2 1}^a E_{3 1}^b E_{3 1}^c \cdot m
- = (\lambda - a (\alpha_1 - \alpha_2)
- - b (\alpha_1 - \alpha_3)
- - c (\alpha_2 - \alpha_3))(H)
+ = (\lambda - a (\epsilon_1 - \epsilon_2)
+ - b (\epsilon_1 - \epsilon_3)
+ - c (\epsilon_2 - \epsilon_3))(H)
E_{2 1}^a E_{3 1}^b E_{3 1}^c \cdot m
\in N
\]
@@ -1050,45 +1050,45 @@ Specifically\dots
& = E_{2 1} ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2})
E_{2 1}^{a - 2} E_{3 1}^b E_{3 2}^c \cdot m \\
& \phantom{=} \; +
- (\lambda - (a - 1) (\alpha_1 - \alpha_2)
- - b (\alpha_1 - \alpha_3)
- - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+ (\lambda - (a - 1) (\epsilon_1 - \epsilon_2)
+ - b (\epsilon_1 - \epsilon_3)
+ - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}])
E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c \cdot m \\
& = E_{2 1}^2 ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2})
E_{2 1}^{a - 3} E_{3 1}^b E_{3 2}^c \cdot m \\
& \phantom{=} \; +
- (\lambda - (a - 1) (\alpha_1 - \alpha_2)
- - b (\alpha_1 - \alpha_3)
- - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+ (\lambda - (a - 1) (\epsilon_1 - \epsilon_2)
+ - b (\epsilon_1 - \epsilon_3)
+ - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}])
E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c \cdot m \\
& \phantom{=} \; +
- (\lambda - (a - 2) (\alpha_1 - \alpha_2)
- - b (\alpha_1 - \alpha_3)
- - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+ (\lambda - (a - 2) (\epsilon_1 - \epsilon_2)
+ - b (\epsilon_1 - \epsilon_3)
+ - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}])
E_{2 1}^{a - 2} E_{3 1}^b E_{3 2}^c \cdot m \\
& \; \; \vdots \\
& = E_{2 1}^a E_{1 2} E_{3 1}^b E_{3 2}^c \cdot m \\
& \phantom{=} \; +
- (\lambda - (a - 1) (\alpha_1 - \alpha_2)
- - b (\alpha_1 - \alpha_3)
- - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+ (\lambda - (a - 1) (\epsilon_1 - \epsilon_2)
+ - b (\epsilon_1 - \epsilon_3)
+ - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}])
E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c \cdot m \\
& \phantom{=} \; +
- (\lambda - (a - 2) (\alpha_1 - \alpha_2)
- - b (\alpha_1 - \alpha_3)
- - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+ (\lambda - (a - 2) (\epsilon_1 - \epsilon_2)
+ - b (\epsilon_1 - \epsilon_3)
+ - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}])
E_{2 1}^{a - 2} E_{3 1}^b E_{3 2}^c \cdot m \\
& \phantom{=} \; \; \, \vdots \\
& \phantom{=} \; +
- (\lambda - (a - a) (\alpha_1 - \alpha_2)
- - b (\alpha_1 - \alpha_3)
- - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+ (\lambda - (a - a) (\epsilon_1 - \epsilon_2)
+ - b (\epsilon_1 - \epsilon_3)
+ - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}])
E_{2 1}^{a - a} E_{3 1}^b E_{3 2}^c \cdot m \\
\end{split}
\]
- Since \((\lambda - (a - k) (\alpha_1 - \alpha_2) - b (\alpha_1 - \alpha_3) -
- c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}]) E_{2 1}^{a - k} E_{3 1}^b
+ Since \((\lambda - (a - k) (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) -
+ c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}]) E_{2 1}^{a - k} E_{3 1}^b
E_{3 2}^c \cdot m \in N\) for all \(k\), it suffices to show \(E_{2 1}^a E_{1
2} E_{3 1}^b E_{3 2}^c \cdot m \in N\). But
\[
@@ -1135,7 +1135,7 @@ Theorem~\ref{thm:sl3-existence-uniqueness}. Moreover, constructing such a
module turns out to be quite simple.
\begin{proof}[Proof of existence]
- Take \(\lambda = k \alpha_1 - \ell \alpha_3 \in P\) with \(k, \ell \ge 0\),
+ Take \(\lambda = k \epsilon_1 - \ell \epsilon_3 \in P\) 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
@@ -1143,7 +1143,7 @@ module turns out to be quite simple.
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
- \(\alpha_1\), \(\alpha_2\) and \(\alpha_3\) respectively. Hence the weight
+ \(\epsilon_1\), \(\epsilon_2\) and \(\epsilon_3\) respectively. Hence the weight
diagram of \(K^3\) is
\begin{center}
\begin{tikzpicture}[scale=2]
@@ -1153,16 +1153,16 @@ module turns out to be quite simple.
\wt[black]{1}{0}
\wt[black]{-1}{1}
\wt[black]{0}{-1}
- \node[right] at \weight{1}{0} {$\alpha_1$};
- \node[above left] at \weight{-1}{1} {$\alpha_2$};
- \node[below left] at \weight{0}{-1} {$\alpha_3$};
+ \node[right] at \weight{1}{0} {$\epsilon_1$};
+ \node[above left] at \weight{-1}{1} {$\epsilon_2$};
+ \node[below left] at \weight{0}{-1} {$\epsilon_3$};
\end{rootSystem}
\end{tikzpicture}
\end{center}
- and \(\alpha_1\) is the highest weight of \(K^3\).
+ and \(\epsilon_1\) is the highest weight of \(K^3\).
On the one hand, if \(\{f_1, f_2, f_3\}\) is the dual basis for \(\{e_1, e_2,
- e_3\}\) then \(H \cdot f_i = - \alpha_i(H) f_i\) for each \(H \in
+ e_3\}\) then \(H \cdot f_i = - \epsilon_i(H) f_i\) for each \(H \in
\mathfrak{h}\), so that the weights of \((K^3)^*\) are precisely the
opposites of the weights of \(K^3\). In other words,
\begin{center}
@@ -1173,13 +1173,13 @@ module turns out to be quite simple.
\wt[black]{-1}{0}
\wt[black]{1}{-1}
\wt[black]{0}{1}
- \node[left] at \weight{-1}{0} {$-\alpha_1$};
- \node[below right] at \weight{1}{-1} {$-\alpha_2$};
- \node[above right] at \weight{0}{1} {$-\alpha_3$};
+ \node[left] at \weight{-1}{0} {$-\epsilon_1$};
+ \node[below right] at \weight{1}{-1} {$-\epsilon_2$};
+ \node[above right] at \weight{0}{1} {$-\epsilon_3$};
\end{rootSystem}
\end{tikzpicture}
\end{center}
- is the weight diagram of \((K^3)^*\) and \(\alpha_3\) is the highest weight
+ is the weight diagram of \((K^3)^*\) and \(\epsilon_3\) is the highest weight
of \((K^3)^*\).
On the other hand if we fix two \(\mathfrak{sl}_3(K)\)-modules \(N\) and
@@ -1197,8 +1197,8 @@ module turns out to be quite simple.
weights of \(N\) with the weights of \(L\).
This implies that the highest weights of \(\operatorname{Sym}^k K^3\) and
- \(\operatorname{Sym}^\ell (K^3)^*\) are \(k \alpha_1\) and \(- \ell
- \alpha_3\) respectively -- with highest weight vectors \(e_1^k\) and
+ \(\operatorname{Sym}^\ell (K^3)^*\) are \(k \epsilon_1\) and \(- \ell
+ \epsilon_3\) 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
\(\lambda = k e_1 - \ell e_3\) -- with highest weight vector \(e_1^k \otimes