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\paragraph{\bf Exercise 2:}  Some things about $\fsl_2$.\\
Recall, $L(d)$ is the irreducible $\fsl_2$-module with dimension $d+1$. 
\begin{enumerate}
\item Calculate (give the matrices for) the adjoint representation of $\fsl_2$ and decompose it into irreducible summands.

\item Let $V=\{u,v\}$ be the standard representation of $\fsl_2$, given by 
$$x = \begin{pmatrix} 0&1\\0&0\end{pmatrix}, \qquad 
y = \begin{pmatrix} 0&0\\1&0\end{pmatrix}, \qquad 
h = \begin{pmatrix} 1&0\\0&-1\end{pmatrix}.$$

\begin{enumerate}
	\item Classify $V$ as an $\fsl_2$-module (as sums of $L(d)$'s). 
	\item Give a basis for $V \otimes V$ and calculate the matrices for the action of $x, y,$ and $h$ on that basis (For  $g \in \fsl_2$, $g$ acts on $a\otimes b$ by\dots). 
	\item Classify $V \otimes V$ and $V^{\otimes 3}$ as $\fsl_2$-modules. 
	\item (Bonus) Provide a general formula for the decomposition of $V^{\otimes k}$. 
\end{enumerate}

\item With $V$ as in the previous part, define the $k$th symmetric sum of $V$
as 
$$\Sym^k(V) = V^{\otimes k} / \<a\otimes b - b \otimes a\> \cong \CC\{u^k, u^{k-1}v, \dots, v^k\} $$
(since $\Sym^k(V)$ is isomorphic to the degree-$k$ homogeneous elements of $\CC[u,v]$). 

\begin{enumerate}
	\item Generally describe the action of $\fsl_2$ on $\Sym^k(V)$. (For  $g \in \fsl_2$, $g$ acts on $u^\ell v^{k-\ell}$ by\dots)
	\item How does $\Sym^k(V)$ decompose into irreducible summands?
	\item Show that for $a \geq b$,
	$$\Sym^a(V) \otimes \Sym^b(V) \cong \bigoplus_{i=0}^{b} \Sym^{a+b-2i}(V).$$
\end{enumerate}
\end{enumerate}

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