were identified with certain one-column tableaux. The finite crystal $B({\Lambda})$ for a dominant weight ${\Lambda}={\Lambda}_{k_1}+\cdots+
{\Lambda}_{k_m}$ with $k_1\ge k_2\ge \cdots \ge k_m$ is isomorphic to the connected component in $B({\Lambda}_{k_1})\otimes \cdots \otimes B({\Lambda}_{k_m})$ which contains the highest weight element $u_{{\Lambda}_{k_1}}\otimes \cdots
\otimes u_{{\Lambda}_{k_m}}$. Combining the two embeddings it follows that $B({\Lambda})$ is isomorphic to a certain connected component in $B({\Lambda}_1)^{\otimes M}$ where $M=k_1+\cdots+k_m$.

For type $A_n$ the elements in $B({\Lambda})$ can be identified with semi-standard Young tableaux of shape ${\lambda}$ where ${\lambda}$ is the partition corresponding to the weight ${\Lambda}$. The paths encountered in section \[sec:HHM\] are sequences of 1’s and 2’s. Viewed as single box Young tableaux over the alphabet $\{1,2\}$, these are exactly the elements of the crystal $B({\Lambda}_1)$ of type $A_1$.

Simple crystals
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Simple crystals were introduced by Akasaka and Kashiwara [@AK]. As will be explained in the next section they have an isomorphism and energy function which are required for the definition of the one-dimensional configuration sums.

Let ${\hat{W}}$ be the Weyl group of the affine Kac–Moody algebra ${\hat{{\mathfrak{g}}}}$ generated by the simple reflections $r_i$ for $i\in I$ defined as $$\label{reflections}
r_i(\beta)=\beta - \langle h_i,\beta\rangle \alpha_i$$ where $\beta$ is a root. Let $B$ be the 