Let be affine finite crystals. Combinatorial is a unique map commuting with Kashiwara operators . If and have a coordinate representation, the image of is written as a set of piecewise linear functions in these coordinates. Following Kirillov's slogan ``Tropical Combinatorics" we tackle this problem of obtaining the explicit formula of by replacing the notions crystal and combinatorical with geometric crystal, introduced by Berenstein and Kazhdan, and tropical .

This is a joint work with A.Kuniba, T.Takagi and Y.Yamada.