lie-algebras-and-their-representations

diff --git a/TODO.md b/TODO.md
@@ -2,8 +2,6 @@
 
 * Comment on the fact that representation theory is larger than the
   representation theory of Lie algebras
-* Comment on the geometric realization of the irreducible representations of
-  sl2
 * Add some comments on how the concept of coherent families is useful to other
   problems too
 

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -736,11 +736,18 @@ concept of a \(\mathcal{U}(\mathfrak{g})\)-module entirely in terms of
   representation}.
 \end{example}
 
+\begin{example}
+  Given a subalgebra \(\mathfrak{g} \subset \mathfrak{gl}_n(K)\), the inclusion
+  \(\mathfrak{g} \to \mathfrak{gl}_n(K)\) endows \(K^n\) with the structure of
+  a representation of \(\mathfrak{g}\), known as \emph{the natural
+  representation of \(\mathfrak{g}\)}.
+\end{example}
+
 It is usual practice to think of a representation \(V\) of \(\mathfrak{g}\) in
 terms of an action of \(\mathfrak{g}\) in a vector space and write simply \(X
 \cdot v\) or \(X v\) for \(\rho(X) v\). For instance, one might say\dots
 
-\begin{example}
+\begin{example}\label{ex:sl2-polynomial-rep}
   The space \(K[x, y]\) is a representation of \(\mathfrak{sl}_2(K)\) with
   \begin{align*}
     e \cdot p & = x \frac{\mathrm{d}}{\mathrm{d}y} p &
@@ -840,6 +847,14 @@ To that end, we define\dots
   i.e. \(X w \in W\) for all \(w \in W\) and \(X \in \mathfrak{g}\).
 \end{definition}
 
+\begin{example}\label{ex:sl2-polynomial-subrep}
+  Let \(K[x, y]\) be the \(\mathfrak{sl}_2(K)\)-module as in
+  example~\ref{ex:sl2-polynomial-rep}. Since \(e\), \(f\) and \(h\) all
+  preserve the degree of monomials, the space \(K_n[x, y] = \bigoplus_{k = 0}^n
+  K x^{n - k} y^k\) of homogenious polynomials of degree \(n\) is a
+  finite-dimensional subrepresentation of \(K[x, y]\).
+\end{example}
+
 \begin{example}
   Given a Lie algebra \(\mathfrak{g}\), a representation \(V\) of
   \(\mathfrak{g}\) and a subrepresentation \(W \subset V\), the space

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -217,38 +217,40 @@ and
 \end{equation}
 
 To conclude our analysis all it's left is to show that for each \(n\) such
-\(V\) does indeed exist and is irreducible. In other words\dots
+\(V\) does indeed exist and is irreducible. Surprinsingly, we have already
+encountered such a \(V\).
 
 \begin{theorem}\label{thm:irr-rep-of-sl2-exists}
-  For each \(n \ge 0\) there exists a (unique) irreducible representation of
+  For each \(n \ge 0\) there exists a unique irreducible representation of
   \(\mathfrak{sl}_2(K)\) whose left-most eigenvalue of \(h\) is \(n\).
 \end{theorem}
 
 \begin{proof}
-  The fact the representation \(V\) from the previous discussion exists is
-  clear from the commutator relations of \(\mathfrak{sl}_2(K)\) -- just look at
-  \(f^k v\) as abstract symbols and impose the action given by
-  (\ref{eq:irr-rep-of-sl2}). Alternatively, one can readily check that if
-  \(K^2\) is the natural representation of \(\mathfrak{sl}_2(K)\), then \(V =
-  \operatorname{Sym}^n K^2\) satisfies the relations of
-  (\ref{eq:irr-rep-of-sl2}). To see that \(V\) is irreducible let \(W\) be a
-  non-zero subrepresentation and take some non-zero \(w \in W\). Suppose \(w =
-  \alpha_0 v + \alpha_1 f v + \cdots + \alpha_n f^n v\) and let \(k\) be the
-  lowest index such that \(\alpha_k \ne 0\), so that
-  \[
-    w = \alpha_k f^k v + \cdots + \alpha_n f^n v
-  \]
-
-  Now given that \(f^m = f^{n + 1}\) annihilates \(v\),
-  \[
-    f w = \alpha_k f^{k + 1} v + \cdots + \alpha_{n - 1} f^n v
-  \]
+  Let \(V = K_n[x, y]\) be as in example~\ref{ex:sl2-polynomial-subrep}. A
+  simple calculation shows \(V_{n - 2 k} = K x^{n - k} y^k\) for \(k = 0,
+  \ldots, n\) and \(V_\lambda = 0\) otherwise. In particular, the right-most
+  eigenvalue of \(V\) is \(n\). Moreover, \(V\) must be irreducible, for if
+  \(V\) could be decomposed as the sum of at least two irreducible
+  representations we would either find an eigenspace of dimension greater than
+  \(1\) or find an eigenvalue whose difference with \(n\) is odd.
+  Alternatively, one can readily check that if \(K^2\) is the natural
+  representation of \(\mathfrak{sl}_2(K)\), then \(V = \operatorname{Sym}^n
+  K^2\) satisfies the relations of (\ref{eq:irr-rep-of-sl2}). Indeed, the map
+  \begin{align*}
+        K_n[x, y] & \to     \operatorname{Sym}^n K^2 \\
+    x^{n - k} y^k & \mapsto e_1^{n - k} \cdot e_2^k
+  \end{align*}
+  is an isomorphism.
 
-  Proceeding inductively we arrive at \(f^{n - k} w = \alpha_k f^n v\), so
-  that \(f^n v \in W\). Hence \(e^i f^n v = \prod_{k = 1}^i k(n + 1 - k) f^{n -
-  i} v \in W\) for all \(i = 1, 2, \ldots, n\). Since \(k \ne 0 \ne n + 1 - k\)
-  for all \(k\) in this range, we can see that \(f^k v \in W\) for all \(k = 0,
-  1, \ldots, n\). In other words, \(W = V\). We are done.
+  As for the uniqueness of \(V\), it suffices to notice that if \(W\) is a
+  finite-dimensional irreducible representation of \(\mathfrak{sl}_2(K)\) with
+  right-most eigenvector \(w\) then relations (\ref{eq:irr-rep-of-sl2}) imply
+  the map
+  \begin{align*}
+        V & \to     W     \\
+    f^k v & \mapsto f^k w
+  \end{align*}
+  is an isomorphism.
 \end{proof}
 
 Our initial gamble of studying the eigenvalues of \(h\) may have seemed