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.