Geometric Robinson-Schensted-Knuth correspondence

We define a geometric RSK correspondence for any semisimple group and any reduced decompositon of element of its Weyl group. This correspondence is a biration map of tori of dimension equal to the length of a reduced decomposition.

For the longest element of the Weyl group, the tropicalization of this map turns out to be the isomorphism between the Lusztig crystal on the canonical basis and the Kashiwara crystal on the dual canonical basis. The geometric corespondence provide us with a transformation of the corresponding superpotentials for geometric crystals.

For the case $G=S{L}_{n+m}$ and the grassmannian permutation, our construction lead to a modified variant of usual RSK being a bijection between non-negative $n\times m$ arrays and pairs of semistandard Young tableaux of equal form (modifiction means that we get a pair $(P^{Sch},Q)$ for an array, where $P^{Sch}$ denotes the tableaux being the Sch\"utzenberger involution to $P$, while usual RSK ends with the pair $(P,Q)$). The geometric RSK, in such a case, transforms the Berenstein-Kazdan potential to the superpotential due to Riesch-Williams.