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\paragraph{\bf Exercise 5:} Some things about weights and representations.
\begin{enumerate}
\item Let $\fg$ be a finite-dimensional complex semisimple Lie algebra.
\begin{enumerate}
\item Show that if $L(\lambda)$ and $L(\mu)$ are highest weight modules (of weights $\lambda$ and $\mu$), show that $L(\lambda) \otimes L(\mu)$ has $L(\lambda+\mu)$ as a submodule with multiplicity 1. (Think about primitive elements)
\item Show that $0$ is a weight of highest weight module $L(\lambda)$ if and only if $\lambda$ is a sum of roots.
\end{enumerate}
\item {\bf Type $A_r$ stuff.} Analyze the standard representation of $\fsl_3$.
\begin{enumerate}
\item What are the primitive elements?
\item What is/are the weight/weights of the action of $\fh$ on the primitive elements (in terms of $\omega_1$ and $\omega_2$)?
\item What is the standard representation isomorphic to (in terms of highest weight modules)?
\item Draw a picture of the weights and verify that the dimension is correct.
\item What is the standard representation (in terms of highest weight modules) of $\fsl_{r+1}$ in general?
\end{enumerate}
\item {\bf Type $C_r$ stuff.}
\begin{enumerate}
\item Give a base for the set of roots of type $C_r$, and calculate the corresponding simple co-roots and fundamental weights.
\item Give two examples of highest weight modules for $C_2$ for which every weight space has multiplicity 1 (and justify how you know every weight space has multiplicity 1).
\end{enumerate}
\end{enumerate}
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