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\paragraph{\bf Exercise 7:} Some things about crystals.\\
\begin{enumerate}
\item Verify that the two formulas for the Weyl denominator $a_\rho$ agree for type $A_2$.
\item Use the three methods from class (Freudenthal's multiplicity formula, the Weyl character formula, and the path model) to calculate the (non-obvious) multiplicities in $L(\omega_2)$ for type $C_2$. Draw the crystal graph for $\cB(\omega_2)$.
\item In type $A_r$, we saw that the finite dimensional modules are indexed by integer partitions of length no more than $r$, that the weight $\omega_1$ corresponds to the integer partition of 1, and that for any $\lambda \in P^+$, the decomposition of $L(\lambda) \otimes L(\omega_1)$ is given by
$$L(\lambda) \otimes L(\omega_1) = \sum_{\mu \in \lambda^+} L(\mu)$$
where $\lambda^+$ is the set of all weights corresponding to partitions obtained by adding a box to the partition corresponding to the weight $\lambda$.
In type $C_r$, there's a similar story. Choose the base
$$B= \{\vep_1 - \vep_2, \vep_2- \vep_3, \dots, \vep_{r-1} - \vep_r , 2 \vep_r\}$$
so that the fundamental weights are given by
$$\omega_i = \vep_1 + \dots + \vep_i \qquad \text{ for } i=1, \dots, r.$$
Then
$$P^+ = \left\{\lambda_1 \vep_1 + \cdots + \lambda_r \vep_r ~\left|~
\begin{matrix} \lambda_i \in \ZZ \\
\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_r \geq 0\end{matrix}\right.\right\}$$
is in bijection with integer partitions of length at most $r$ (with less work than in type $A_r$, even).
Use the path model to determine the decomposition of $L(\lambda) \otimes L(\omega_1)$ in type $C_r$, expressing the answer in terms of partitions.
\end{enumerate}
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