In the paper, 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.

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