that $$D_B(u(B))=0.$$

Finite dimensional affine crystals
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Recently, new families of crystals of the finite dimensional representations of $U_q({\hat{{\mathfrak{g}}}})$ were conjectured [@HKOTY; @K3] where ${\hat{{\mathfrak{g}}}}$ is a untwisted affine Lie algebra.

[@HKOTY; @K3]\[conj:Brs\] For each $r\in J$ and $s\ge1$, there exists an irreducible finite-dimensional integrable $U'_q({\hat{{\mathfrak{g}}}})$-module $W^{(r)}_s$ with simple crystal base $B^{r,s}$ generated a unique extremal vector $u(B^{r,s})$ of weight $s{\Lambda}_r$, and a prescribed $U_q({\mathfrak{g}})$-crystal decomposition of the form $B^{r,s} \cong
B(s{\Lambda}_r)\oplus B$, where $B$ is a direct sum of $U_q({\mathfrak{g}})$-crystals of the form $B({\Lambda})$ where ${\Lambda}$ is a classical dominant weight and $s {\Lambda}_r \vartriangleright {\lambda}$. Here ${\Lambda}'\trianglerighteq{\Lambda}$ if and only if ${\Lambda}'-{\Lambda}\in\bigoplus_{i\in J} {\mathbb{N}}\alpha_i$. Moreover there is a prescribed intrinsic energy function $D=D_{B^{r,s}}:B^{r,s}\rightarrow {\mathbb{Z}}$, that is constant on $J$-components, such that $0=D(u)>D(b)$ where $u$ is the $J$-highest weight vector of weight $s{\Lambda}_r$ in $B^{r,s}$, and $b$ is any element not in the $J$-component of $u$.

Combinatorial $R$-matrix and energy function
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Suppose $B_1$ and $B_2$ are simple $U'_q({\mathfrak{g}})$-crystals. Then there is a unique isomorphism of $U'_q({\mathfrak{g}})$-crystals $\sigma:B_2\otimes B_1\rightarrow B_1\otimes B_2$. There is also a function $H=H_{B_2,B_1}:B_2\otimes B_1\rightarrow{\mathbb{Z}}$, called local energy function which is unique up to global additive constant, such that for all $b_2\otimes b_1\in B_2\otimes B_1$ with 