is $\rho=(n,n-1,\dotsc,1,0)-\frac{n}{2}e$. The scalar product is the standard one: $(\alpha|\beta)=\alpha\cdot\beta$.

For ${\mathfrak{g}}=C_n$ the simple roots are given by the short roots $\alpha_i=\epsilon_i-\epsilon_{i+1}$ for $1\le i<n$ and the long root $\alpha_n=2\epsilon_n$ where $\epsilon_i$ is the $i$-th unit vector in ${\mathbb{R}}^n$. The fundamental weights are ${\Lambda}_i=\epsilon_1+\cdots+\epsilon_i$ for $i\in J$. Half the sum of all positive roots is $\rho=(n,n-1,\ldots,1)$. The bilinear form is given by $(\alpha|\beta)=\alpha\cdot \beta/2$.

For both type $A_n$ and $C_n$ we will identify dominant weights, that is weights of the form ${\Lambda}={\Lambda}_{k_1}+\cdots+{\Lambda}_{k_m}$, with partitions. A partition ${\lambda}=({\lambda}_1,\ldots,{\lambda}_{n+1})$ (with ${\lambda}_{n+1}=0$ for type $C_n$) corresponds to the weight ${\Lambda}=\sum_{i\in J}({\lambda}_i-{\lambda}_{i+1}){\Lambda}_i$.

For each dominant weight $\Lambda$ there is a finite classical crystal $B(\Lambda)$ [@KN]. The crystals $B(\Lambda_1)$ for type $A_n$ and $C_n$ are given in Table \[table: classical crystals\] where the arrow $\stackrel{i}{\longrightarrow}$ stands for $f_i$.

   $A_n$  
  ------- --
   $C_n$  

  : \[table: classical crystals\]Classical crystals $B({\Lambda}_1)$

The finite crystals $B({\Lambda}_k)$ for $k\in J$ can be obtained in the following way. Let $u_{{\Lambda}_k}=
\raisebox{-0.14cm}{\scalebox{0.8}{\begin{picture}(14,14)(0,-7)
\BBoxc(7,0)(14,14)\Text(7,0)[]{$k$}\end{picture}}}
\otimes \cdots \otimes
\raisebox{-0.14cm}{\scalebox{0.8}{\begin{picture}(14,14)(0,-7)
\BBoxc(7,0)(14,14)\Text(7,0)[]{$2$}\end{picture}}}
\otimes
\raisebox{-0.14cm}{\scalebox{0.8}{\begin{picture}(14,14)(0,-7)
\BBoxc(7,0)(14,14)\Text(7,0)[]{$1$}\end{picture}}}$ be the unique highest weight vector of weight ${\Lambda}_k$ in $B({\Lambda}_1)^{\otimes k}$. Then $B({\Lambda}_k)$ is the connected component of $B({\Lambda}_1)^{\otimes k}$ containing $u_{{\Lambda}_k}$. In [@KN] the elements in this connected component 