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Affine Geometric Crystal of $A^{(1)}_n$ and Limit of Kirillov-Reshetikhin Perfect Crystals

Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and ${\mathfrak g}^L$ be its Langlands dual. It is conjectured by Kashiwara et al.([16]) that for each $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for ${\mathfrak g}^L$. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra ${\mathfrak g}= A^{(1)}_n$ for each Dynkin index $k\in I\setminus\{0\}$ and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for $A^{(1)}_n$ given by Okado et al.([29]). In the process we develop and use some lattice-path combinatorics.