Kirillov-Reshetikhin crystals, energy function and the combinatorial R-matrix

We study the polytope model for the affine type $A$ Kirillov-Reshetikhin crystals and prove that the action of the affine Kashiwara operators can be described in a remarkable simple way. Moreover, we investigate the combinatorial $R$-matrix on a tensor product of polytopes and characterize the map explicitly on the highest weight elements. We further give a formula for the local energy function and provide an alternative proof for the perfectness. We determine for any dominant highest weight element $\Lambda$ of level $\ell$ the elements $b_{\Lambda}, b^{\Lambda}$ involved in the definition of perfect crystals and give an explicit description of the ground-state path in the tensor product of polytopes.