Affine geometric crystal of An(1) and limit of Kirillov-Reshetikhin perfect crystals

Let g be an affine Lie algebra with index set I={0,1,2,⋯,n} and gL be its Langlands dual. It is conjectured in [16] that for each k∈I∖{0} the affine Lie algebra g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for gL. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra g=An(1) for each Dynkin index k∈I∖{0} and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for An(1) given in [24]. In the process we develop and use some lattice-path combinatorics.