TITLE: RSK correspondence
QUESTION [8 upvotes]: Up to now, what are the difference ways we know to define RSK correspondence? I already know:

By insertion and recording tableau.
Ball construction or Viennot's geometric construction.
Growth diagram proposed by Sergey Fomin.

Do you know other models?

REPLY [4 votes]: Here, slightly edited, is the first paragraph of Steinberg's paper, An occurrence of the Robinson–Schensted correspondence.

Let $V$ be an $n$-dimensional vector space over an infinite field, $\mathscr F$ the flag manifold of $V$, $u$ a unipotent transformation of $V$, and $\lambda$ the type of $u$, a partition of $n$ whose parts are the sizes of the Jordan blocks for $u$. … The components of $\mathscr F_u$, the variety of flags fixed by $u$, correspond naturally to the standard tableaux of shape $\lambda$. The purpose of this note is to show that the "relative position" of any two components of $\mathscr F_u$ (in general an element of the Weyl group, in the present case an element of $S_n$) is given, in terms of the corresponding tableaux, by the Robinson–Schensted correspondence.