lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -169,11 +169,11 @@ We can find an orthonormal basis \(\{\epsilon_1, \ldots, \epsilon_n\}\) for
 % TODOO: Add notes about this basis beforehand
 Take the standard Cartan subalgebra \(\mathfrak{h} = \{ X \in \mathfrak{sl}_{n
 + 1}(K) : X \ \text{is diagonal}\}\) of \(\mathfrak{sl}_{n + 1}(K)\) as in
-Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\alpha_i,
-\ldots, \alpha_{n + 1} \in \mathfrak{h}^*\) such that \(\alpha_i(H)\) is the
+Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
+\ldots, \epsilon_{n + 1} \in \mathfrak{h}^*\) such that \(\epsilon_i(H)\) is the
 \(i\)-th entry of the diagonal of \(H\). Let \(\Sigma = \{ \beta_1, \ldots,
-\beta_n \}\) be the basis for \(\Delta\) given by \(\beta_i = \alpha_i -
-\alpha_{i + 1}\).
+\beta_n \}\) be the basis for \(\Delta\) given by \(\beta_i = \epsilon_i -
+\epsilon_{i + 1}\).
 
 % TODO: Add some comments on the proof of this: while the proof that these
 % conditions are necessary is a purely combinatorial affair, the proof of the
@@ -216,9 +216,9 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\alpha_i,
         \lambda &
         \mapsto
         (
-          \kappa(\alpha_1, \lambda + \rho),
+          \kappa(\epsilon_1, \lambda + \rho),
           \cdots,
-          \kappa(\alpha_{n + 1}, \lambda + \rho)
+          \kappa(\epsilon_{n + 1}, \lambda + \rho)
         )
   \end{align*}
   is \(W\)-equivariant bijection onto the space of all \(\mathfrak{sl}_{n +

diff --git a/sections/fin-dim-simple.tex b/sections/fin-dim-simple.tex
@@ -469,7 +469,7 @@ restrictions on the weights of \(M\). Namely, if \(\lambda\) is a weight,
 Proposition~\ref{thm:weights-fit-in-weight-lattice} is clearly analogous to
 Corollary~\ref{thm:sl3-weights-fit-in-weight-lattice}. In fact, the weight
 lattice of \(\mathfrak{sl}_3(K)\) -- as in Definition~\ref{def:weight-lattice}
--- is precisely \(\mathbb{Z} \langle \alpha_1, \alpha_2, \alpha_3 \rangle\). To
+-- is precisely \(\mathbb{Z} \langle \epsilon_1, \epsilon_2, \epsilon_3 \rangle\). To
 proceed further, we would like to take \emph{the highest weight of \(M\)} as in
 section~\ref{sec:sl3-reps}, but the meaning of \emph{highest} is again unclear
 in this situation. We could simply fix a linear function \(\mathbb{Q} P \to
@@ -600,7 +600,7 @@ This has a number of important consequences. For instance\dots
 This is entirely analogous to the situation of \(\mathfrak{sl}_3(K)\), where we
 found that the weights of the simple \(\mathfrak{sl}_3(K)\)-modules formed
 continuous strings symmetric with respect to the lines \(K \alpha\) with
-\(\kappa(\alpha_i - \alpha_j, \alpha) = 0\). As in the case of
+\(\kappa(\epsilon_i - \epsilon_j, \alpha) = 0\). As in the case of
 \(\mathfrak{sl}_3(K)\), the same class of arguments leads us to the
 conclusion\dots
 

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