Realization of affine type A Kirillov–Reshetikhin crystals via polytopes

On the polytope defined by Feigin, Fourier and Littelmann, associated to any highest weight corresponding to a rectangular partition, we define a crystal structure of type AnAn. We show that this crystal is isomorphic to the one obtained from Kashiwaraʼs crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov–Reshetikhin crystals for the affine type An(1) via polytopes.