At least one perfect crystal for each integrable $U_q({\hat{{\mathfrak{g}}}})$-module for ${\hat{{\mathfrak{g}}}}=A_n^{(1)}, B_n^{(1)},
C_n^{(1)}, D_n^{(1)}, A_n^{(2)}$ and $D_{n+1}^{(2)}$ was given in [@KKMMNN2].

Kashiwara and Nakashima [@KN] constructed the finite $U_q({\mathfrak{g}})$-crystals for ${\mathfrak{g}}=A_n$, $B_n$, $C_n$ and $D_n$ explicitly in terms of tableaux. The cases ${\mathfrak{g}}=A_n$ and $C_n$ are discussed in more detail in the next subsection.

Finite crystals of type $A_n$ and $C_n$ {#sec:finite AC}
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The finite crystals associated with ${\mathfrak{g}}=A_n$ and $C_n$ are presented in more detail since they will be our main examples throughout the paper. Let $\Lambda_i$ for $i\in J$ be the fundamental weights of ${\mathfrak{g}}$. For later purposes it will be convenient to give the root and weight structure of $A_n$ and $C_n$ explicitly. Let $(\cdot|\cdot)$ be the standard bilinear form normalized such that $(\alpha_i\mid\alpha_i)=2$ for the long roots $\alpha_i$.

\[ex:roots\] For either ${\mathfrak{g}}=A_n$ or ${\mathfrak{g}}=C_n$ the index set for the simple roots is $J=\{1,2,\dotsc,n\}$.

For ${\mathfrak{g}}=A_n$ the weight lattice is embedded in the subspace of ${\mathbb{R}}^{n+1}$ orthogonal to the vector $e=\sum_{j=1}^{n+1} \epsilon_j$, where $\epsilon_i$ is the $i$-th standard basis vector of ${\mathbb{R}}^{n+1}$. The simple roots are $\alpha_i=\epsilon_i-\epsilon_{i+1}$ for all $i\in J$. The fundamental weights are ${\Lambda}_i=\sum_{j=1}^i \epsilon_j
-\frac{i}{n+1}e$ for $i\in J$. The half-sum of positive roots 