In the [paper](http://arxiv.org/abs/1210.5126v2), geometric RSK correspondence is given by 
$$
\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \left( \begin{matrix} \frac{bc}{b+c} & ab \\ ac & \frac{ad}{b+c} \end{matrix} \right).
$$
How this map relates to the classical RSK correspondence? Thank you very much.


Edit: the map is
$$
\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \left( \begin{matrix} \frac{bc}{b+c} & ab \\ ac & ad(b+c) \end{matrix} \right).
$$



Geometric RSK correspondece and classical RSK correspondence
I have some notes that hopefully explain some of this: <http://www-users.math.umn.edu/~shopkins/docs/rsk.pdf>.


By the way, I believe you got the map slightly wrong. It should be:
$$
\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \left( \begin{matrix} \frac{bc}{b+c} & ab \\ ac & ad(b+c) \end{matrix} \right).
$$
See Example 7 from the above linked notes to see why the tropicalized version of this map
$$
\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \left( \begin{matrix} \mathrm{min}(b,c) & a+b \\ a+c & a+d+\mathrm{max}(b,c) \end{matrix} \right)
$$
is classical RSK in disguise. I should say I learned all of this material from Alex Postnikov.


