Tropical Robinson-Schensted-Knuth correspondence and birational Weyl   group actions
  By using an elementary matrix approach, based on the technique of discrete Toda equation, we construct subtraction-free rational and piecewise linear transformations associated with various combinatorial algorithms, including the RSK correspondence. We also investigate birational Weyl group actions related to those algorithms. 
Masatoshi Noumi and Yasuhiko Yamada