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

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  • $\begingroup$ Related mathoverflow.net/questions/97899 $\endgroup$
    – David E Speyer
    Commented Jun 19, 2014 at 13:49
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    $\begingroup$ A note -- if you're linking to arXiv, it's better to link to the abstract (arxiv.org/abs/1210.5126v2) rather than directly to the PDF. From the abstract, one can easily click through to the PDF; not so the reverse. And the abstract allows one to do things like see different versions of the paper, search for other things by the same authors, etc. Thank you! $\endgroup$
    – Harry Altman
    Commented Jun 19, 2014 at 19:51
  • $\begingroup$ @HarryAltman, thank you very much. I have edited the post. $\endgroup$
    – Jianrong Li
    Commented Jun 20, 2014 at 6:08

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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.

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