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