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