diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -480,17 +480,171 @@ representation of \(\mathfrak{sl}_3(K)\) deserve some special attention.
Theorem~\ref{thm:sl3-weights-congruent-mod-root} can thus be restated as\dots
+\begin{definition}
+ The lattice \(Q = \ZZ \langle \alpha_i - \alpha_j : i, j = 1, 2, 3 \rangle\)
+ is called \emph{the root lattice of \(\mathfrak{sl}_3(K)\)}.
+\end{definition}
+
\begin{corollary}
The weights of an irreducible representation \(V\) of \(\mathfrak{sl}_3(K)\)
- are all congruent module the lattice \(Q\) generated by the roots \(\alpha_i
- - \alpha_j\) of \(\mathfrak{sl}_3(K)\).
+ are all congruent module the root lattice \(Q\). In other words, the weights
+ of \(V\) all lie in a single \(Q\)-coset of \(t \in
+ \mfrac{\mathfrak{h}^*}{Q}\).
\end{corollary}
+At this point we could keep playing the tedious game of reproducing the
+arguments from the previous section in the context of \(\mathfrak{sl}_3(K)\).
+However, it is more profitable to use our knowlage of the representations of
+\(\mathfrak{sl}_2(K)\) instead. Notice that the canonical inclusion
+\(\mathfrak{gl}_2(K) \to \mathfrak{gl}_3(K)\) -- as described in
+example~\ref{ex:gln-inclusions} -- restricts to an injective homomorphism
+\(\mathfrak{sl}_2(K) \to \mathfrak{sl}_3(K)\). In other words,
+\(\mathfrak{sl}_2(K)\) is isomorphic to the image \(\mathfrak{s}_{1 2} = K
+\langle E_{1 2}, E_{2 1}, [E_{1 2}, E_{2 1}] \rangle \subset
+\mathfrak{sl}_3(K)\) of the inclusion \(\mathfrak{sl}_2(K) \to
+\mathfrak{sl}_3(K)\). We may thus consider regard \(V\) as an
+\(\mathfrak{sl}_2(K)\)-module by restricting to \(\mathfrak{s}_{1 2}\).
+
+Our first observation is that, since the root spaces act by translation, the
+subspace
+\[
+ \bigoplus_{k \in \mathbb{Z}} V_{\lambda + k (\alpha_1 - \alpha_2)},
+\]
+must be invariant under the action of \(E_{1 2}\) and \(E_{2 1}\) for all
+\(\lambda \in \mathfrak{h}^*\). This goes to show \(\bigoplus_k V_{\lambda + k
+(\alpha_1 - \alpha_2)}\) 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 \(V_{\lambda + k (\alpha_2 - \alpha_1)}\).
+
+Visually,
+\begin{center}
+ \begin{tikzpicture}
+ \begin{rootSystem}{A}
+ \node at \weight{-4}{2} (l) {};
+ \node at \weight{-2}{1} (a) {};
+ \node at \weight{0}{0} (b) {};
+ \node at \weight{2}{-1} (c) {};
+ \node at \weight{4}{-2} (r) {};
+ \draw \weight{-3}{1.5} -- \weight{3}{-1.5};
+ \draw[dotted] \weight{-3}{1.5} -- (l);
+ \draw[dotted] \weight{3}{-1.5} -- (r);
+ \foreach \i in {-1, 0, 1}{\wt[black]{-2*\i}{\i}}
+ \draw[-latex] (l) to[bend left=40] (a);
+ \draw[-latex] (a) to[bend left=40] (b);
+ \draw[-latex] (b) to[bend left=40] (c);
+ \draw[-latex] (c) to[bend left=40] (r);
+ \draw[-latex] (r) to[bend left=40] (c);
+ \draw[-latex] (c) to[bend left=40] (b);
+ \draw[-latex] (b) to[bend left=40] (a);
+ \draw[-latex] (a) to[bend left=40] (l);
+ \node[above right] at (b) {\small\(\lambda\)};
+ \node[above right=2pt] at \weight{-3}{1.5} {\small\(E_{1 2}\)};
+ \node[below left=2pt] at \weight{-3}{1.5} {\small\(E_{2 1}\)};
+ \end{rootSystem}
+ \end{tikzpicture}
+\end{center}
+
+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
+ \[
+ W = \bigoplus_{k \in \mathbb{Z}} V_{\lambda + k (\alpha_i - \alpha_j)}
+ \]
+ is invariant under the action of \(\mathfrak{s}_{i j}\) and
+ \[
+ V_{\lambda + k (\alpha_i - \alpha_j)}
+ = W_{\lambda([E_{i j}, E_{j i}]) - 2k}
+ \]
+\end{proposition}
+
+\begin{proof}
+ For the first claim, it suffices to notice the map
+ \begin{align*}
+ \mathfrak{sl}_2(K) & \to \mathfrak{s}_{i j} \\
+ e & \mapsto E_{i j} \\
+ f & \mapsto E_{j i} \\
+ h & \mapsto [E_{i j}, E_{j i}]
+ \end{align*}
+ is an isomorphism.
+
+ 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 \(E_{j i} v \in V_{\lambda + (k - 1) (\alpha_i -
+ \alpha_j)}\). Moreover,
+ \[
+ (\lambda + k (\alpha_i - \alpha_j))(H)
+ = \lambda([E_{i j}, E_{j i}]) + k (-1 - 1)
+ = \lambda([E_{i j}, E_{j i}]) - 2 k,
+ \]
+ which goes to show \(V_{\lambda + k (\alpha_i - \alpha_j)} \subset
+ W_{\lambda([E_{i j}, E_{j i}]) - 2k}\). On the other hand, if we suppose \(0
+ < \dim V_{\lambda + k (\alpha_i - \alpha_j)} < \dim W_{\lambda([E_{i j}, E_{j
+ i}]) - 2 k}\) for some \(k\) we arrive at
+ \[
+ \dim W
+ = \sum_k \dim V_{\lambda + k (\alpha_i - \alpha_j)}
+ < \sum_k \dim W_{\lambda([E_{i j}, E_{j i}]) - 2k}
+ = \dim W,
+ \]
+ a contradiction.
+\end{proof}
+
+As a first consequence of this, we show\dots
+
\begin{definition}
- The lattice \(Q = \ZZ \langle \alpha_i - \alpha_j : i, j = 1, 2, 3 \rangle\)
- is called \emph{the root lattice of \(\mathfrak{sl}_3(K)\)}.
+ The lattice \(P = \ZZ \alpha_1 \oplus \ZZ \alpha_2 \oplus \ZZ \alpha_3\) is
+ called \emph{the weight lattice of \(\mathfrak{sl}_3(K)\)}.
\end{definition}
+\begin{corollary}
+ Every weight \(\lambda\) of \(V\) lies in the weight lattice \(\lambda\).
+\end{corollary}
+
+\begin{proof}
+ It suffices to note \(\lambda([E_{i j}, E_{j i}])\) is an eigenvalue of \(h\)
+ in a finite-dimensional representation of \(\mathfrak{sl}_2(K)\), so it must
+ be an integer. Now since
+ \[
+ \lambda
+ \begin{pmatrix}
+ a & 0 & 0 \\
+ 0 & b & 0 \\
+ 0 & 0 & -a -b
+ \end{pmatrix}
+ =
+ \lambda
+ \begin{pmatrix}
+ a & 0 & 0 \\
+ 0 & 0 & 0 \\
+ 0 & 0 & -a
+ \end{pmatrix}
+ +
+ \lambda
+ \begin{pmatrix}
+ 0 & 0 & 0 \\
+ 0 & b & 0 \\
+ 0 & 0 & -b
+ \end{pmatrix}
+ =
+ a \lambda([E_{1 3}, E_{3 1}]) + b \lambda([E_{2 3}, E_{3 2}]),
+ \]
+ which is to say \(\lambda = \lambda([E_{1 3}, E_{3 1}]) \alpha_1 +
+ \lambda([E_{2 3}, E_{3 2}]) \alpha_2 \in P\).
+\end{proof}
+
+There's a clear parallel between the case of \(\mathfrak{sl}_3(K)\) and that of
+\(\mathfrak{sl}_2(K)\), where we observed that the weights all lied in the
+lattice \(P = \ZZ\) and were congruent modulo the sublattice \(Q = 2 \ZZ\).
+Among other things, this last result goes to show that the diagrams we've been
+drawing are in fact consistant with the theory we've developed. Namely, since
+all weights lie in the rational span of \(\{\alpha_1, \alpha_2, \alpha_3\}\) in
+\(\mathfrak{h}^*\) we may as well draw them in the Cartesia plane.
+
To proceed we once more refer to the previously established framework: next we
saw that the eigenvalues of \(h\) formed an unbroken string of integers
symmetric around \(0\). To prove this we analyzed the right-most eigenvalue of
@@ -517,17 +671,16 @@ and let \(\lambda\) be the weight lying the furthest in this direction.
Its easy to see what we mean intuitively by looking at the previous picture,
but its precise meaning is still allusive. Formally this means we'll choose a
-linear functional \(f : \mathfrak{h}^* \to \QQ\) and pick the weight that
-maximizes \(f\). To avoid any ambiguity we should choose the direction of a
-line irrational with respect to the root lattice \(Q\). For instance if we
-choose the direction of \(\alpha_1 - \alpha_3\) and let \(f\) be the rational
-projection \(Q \to \QQ \langle \alpha_1 - \alpha_3 \rangle \cong \QQ\) then
-\(\alpha_1 - 2 \alpha_2 + \alpha_3 \in Q\) lies in \(\ker f\), so that if a
-weight \(\lambda\) maximizes \(f\) then the translation of \(\lambda\) by any
-multiple of \(\alpha_1 - 2 \alpha_2 + \alpha_3\) must also do so. In others
-words, if the direction we choose is parallel to a vector lying in \(Q\) then
-there may be multiple choices the ``weight lying the furthest'' along this
-direction.
+linear functional \(f : \QQ P \to \QQ\) and pick the weight that maximizes
+\(f\). To avoid any ambiguity we should choose the direction of a line
+irrational with respect to the root lattice \(Q\). For instance if we choose
+the direction of \(\alpha_1 - \alpha_3\) and let \(f\) be the projection \(\QQ
+P \to \QQ \langle \alpha_1 - \alpha_3 \rangle \cong \QQ\) then \(\alpha_1 - 2
+\alpha_2 + \alpha_3 \in Q\) lies in \(\ker f\), so that if a weight \(\lambda\)
+maximizes \(f\) then the translation of \(\lambda\) by any multiple of
+\(\alpha_1 - 2 \alpha_2 + \alpha_3\) must also do so. In others words, if the
+direction we choose is parallel to a vector lying in \(Q\) then there may be
+multiple choices the ``weight lying the furthest'' along this direction.
\begin{definition}
We say that a root \(\alpha\) is positive if \(f(\alpha) > 0\) -- i.e. if it
@@ -584,23 +737,14 @@ implemented for \(\mathfrak{sl}_3(K)\) -- and indeed that's what we'll do later
down the line -- but instead we would like to focus on the problem of finding
the weights of \(V\) for the moment.
-% TODO: Move this to before showing V is a highest weight module
-% TODO: This would allow us to justify drawing the weight diagrams in the real
-% plane as opposed to some abstract 4-dimensional space
We'll start out by trying to understand the weights in the boundary of
-\(\frac{1}{3}\)-plane previously drawn. Since the root spaces act by
-translation, the action of \(E_{2 1}\) in \(V_\lambda\) will span a subspace
-\[
- W = \bigoplus_k V_{\lambda + k (\alpha_2 - \alpha_1)},
-\]
-and by the same token \(W\) must be invariant under the action of \(E_{1 2}\).
-
-To draw a familiar picture
+\(\frac{1}{3}\)-plane previously drawn. As we've just seen, we can get to other
+weight spaces from \(V_\lambda\) by successively applying \(E_{1 2}\).
\begin{center}
\begin{tikzpicture}
\begin{rootSystem}{A}
- \node at \weight{3}{1} (a) {};
- \node at \weight{1}{2} (b) {};
+ \node at \weight{3}{1} (a) {};
+ \node at \weight{1}{2} (b) {};
\node at \weight{-1}{3} (c) {};
\node at \weight{-3}{4} (d) {};
\node at \weight{-5}{5} (e) {};
@@ -620,38 +764,16 @@ To draw a familiar picture
\end{tikzpicture}
\end{center}
-What's remarkable about all this is the fact that the subalgebra spanned by
-\(E_{1 2}\), \(E_{2 1}\) and \(H = [E_{1 2}, E_{2 1}]\) is isomorphic to
-\(\mathfrak{sl}_2(K)\) via
-\begin{align*}
- E_{2 1} & \mapsto e &
- E_{1 2} & \mapsto f &
- H & \mapsto h
-\end{align*}
-
-In other words, \(W\) is a representation of \(\mathfrak{sl}_2(K)\). Even more
-so, we claim
-\[
- V_{\lambda + k (\alpha_2 - \alpha_1)} = W_{\lambda(H) - 2k}
-\]
-
-Indeed, \(V_{\lambda + k (\alpha_2 - \alpha_1)} \subset W_{\lambda(H) - 2k}\)
-since \((\lambda + k (\alpha_2 - \alpha_1))(H) = \lambda(H) + k (-1 - 1) =
-\lambda(H) - 2 k\). On the other hand, if we suppose \(0 < \dim V_{\lambda + k
-(\alpha_2 - \alpha_1)} < \dim W_{\lambda(H) - 2 k}\) for some \(k\) we arrive
-at
-\[
- \dim W
- = \sum_k \dim V_{\lambda + k (\alpha_2 - \alpha_1)}
- < \sum_k \dim W_{\lambda(H) - 2k}
- = \dim W,
-\]
-a contradiction.
-
-There are a number of important consequences to this, of the first being that
-the weights of \(V\) appearing on \(W\) must be symmetric with respect to the
-the line \(B(\alpha_1 - \alpha_2, \alpha) = 0\). The picture is
-thus
+Notice that \(\lambda([E_{1 2}, E_{2 1}]) \in \mathbb{Z}\) is the right-most
+eigenvalue of the \(\mathfrak{sl}_2(K)\)-module \(\bigoplus_{k \in \mathbb{Z}}
+V_{\lambda + k (\alpha_i - \alpha_j)}\). 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 \(h\) in
+\(\bigoplus_{k \in \mathbb{N}} V_{\lambda + k (\alpha_1 - \alpha_2)}\) is
+\(V_{\lambda + k (\alpha_1 - \alpha_2)}\), the weights of \(V\) appearing the
+string \(\lambda, \lambda + (\alpha_1 - \alpha_2), \ldots, \lambda + k
+(\alpha_1 - \alpha_2), \ldots\) must be symmetric with respect to the the line
+\(B(\alpha_1 - \alpha_2, \alpha) = 0\). The picture is thus
\begin{center}
\begin{tikzpicture}
\AutoSizeWeightLatticefalse
@@ -669,12 +791,10 @@ thus
\end{tikzpicture}
\end{center}
-Notice we could apply this same argument to the subspace \(\bigoplus_k
-V_{\lambda + k (\alpha_3 - \alpha_2)}\): this subspace is invariant under the
-action of the subalgebra spanned by \(E_{2 3}\), \(E_{3 2}\) and \([E_{2 3},
-E_{3 2}]\), which is again isomorphic to \(\mathfrak{sl}_2(K)\), so that the
-weights in this subspace must be symmetric with respect to the line
-\(B(\alpha_3 - \alpha_2, \alpha) = 0\). The picture is now
+We could apply this same argument to the subspace \(\bigoplus_k V_{\lambda + k
+(\alpha_3 - \alpha_2)}\), so that the weights in this subspace must be
+symmetric with respect to the line \(B(\alpha_3 - \alpha_2, \alpha) = 0\). The
+picture is now
\begin{center}
\begin{tikzpicture}
\AutoSizeWeightLatticefalse
@@ -696,15 +816,8 @@ weights in this subspace must be symmetric with respect to the line
\end{tikzpicture}
\end{center}
-In general, given a weight \(\mu\), the space
-\[
- \bigoplus_k V_{\mu + k (\alpha_i - \alpha_j)}
-\]
-is invariant under the action of the subalgebra \(\mathfrak{s}_{\alpha_i -
-\alpha_j} = K \langle E_{i j}, E_{j i}, [E_{i j}, E_{j i}] \rangle\), which is
-once more isomorphic to \(\mathfrak{sl}_2(K)\), and again the weight spaces in
-this string match precisely the eigenvalues of \(h\). Needless to say, we could
-keep applying this method to the weights at the ends of our string, arriving at
+Needless to say, we could keep applying this method to the weights at the ends
+of our string, arriving at
\begin{center}
\begin{tikzpicture}
\AutoSizeWeightLatticefalse
@@ -814,46 +927,6 @@ must also be weights of \(V\). The final picture is thus
\end{tikzpicture}
\end{center}
-% TODO: Move this to just after the discussion on the distinguished subalgebras
-% TODO: This would allow us to justify the fact we've been drawing the highest
-% weight of V as an element of the weight lattice
-Another important consequence of our analysis is the fact that \(\lambda\) lies
-in the lattice \(P\) generated by \(\alpha_1\), \(\alpha_2\) and \(\alpha_3\).
-Indeed, \(\lambda([E_{i j}, E_{j i}])\) is an eigenvalue of \(h\) in a
-representation of \(\mathfrak{sl}_2(K)\), so it must be an integer. Now since
-\[
- \lambda
- \begin{pmatrix}
- a & 0 & 0 \\
- 0 & b & 0 \\
- 0 & 0 & -a -b
- \end{pmatrix}
- =
- \lambda
- \begin{pmatrix}
- a & 0 & 0 \\
- 0 & 0 & 0 \\
- 0 & 0 & -a
- \end{pmatrix}
- +
- \lambda
- \begin{pmatrix}
- 0 & 0 & 0 \\
- 0 & b & 0 \\
- 0 & 0 & -b
- \end{pmatrix}
- =
- a \lambda([E_{1 3}, E_{3 1}]) + b \lambda([E_{2 3}, E_{3 2}]),
-\]
-which is to say \(\lambda = \lambda([E_{1 3}, E_{3 1}]) \alpha_1 +
-\lambda([E_{2 3}, E_{3 2}]) \alpha_2\), we can see that \(\lambda \in
-P\).
-
-\begin{definition}
- The lattice \(P = \ZZ \alpha_1 \oplus \ZZ \alpha_2 \oplus \ZZ \alpha_3\) is
- called \emph{the weight lattice of \(\mathfrak{sl}_3(K)\)}.
-\end{definition}
-
Finally\dots
\begin{theorem}\label{thm:sl3-irr-weights-class}
@@ -863,9 +936,6 @@ Finally\dots
reflections across the lines \(B(\alpha_i - \alpha_j, \alpha) = 0\).
\end{theorem}
-Once more there's a clear parallel between the case of \(\mathfrak{sl}_3(K)\)
-and that of \(\mathfrak{sl}_2(K)\), where we observed that the weights all lied
-in the lattice \(P = \ZZ\) and were congruent modulo the lattice \(Q = 2 \ZZ\).
Having found all of the weights of \(V\), the only thing we're missing is an
existence and uniqueness theorem analogous to
theorem~\ref{thm:sl2-exist-unique}. In other words, our next goal is