# RSK correspondence

Up to now, what are the difference ways we know to define RSK correspondence? I already know:

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

Do you know other models?

• There is a very enlightening description of RSK in terms of piecewise-linear maps that goes back to Igor Pak: eudml.org/doc/121696 – Sam Hopkins Feb 3 at 17:39
• See also the presentation of Robin Sulzgruber: dx.doi.org/10.4310/JOC.2020.v11.n2.a3 – Sam Hopkins Feb 3 at 17:41
• There is also a direct connection of RSK to important topics in representation theory like Hecke algebras, Kazhdan-Lusztig theory, Springer fibers, etc. and that perspective is probably the most 'canonical'. But it sounds like you are interested in combinatorial constructions. – Sam Hopkins Feb 3 at 17:52
• Do you know about the relation between KL cells and Knuth equivalence? See e.g. arxiv.org/abs/math/9910117 for the basics along these lines. – Sam Hopkins Feb 3 at 18:06
• Greene's theorem describes the shape of the insertion tableau in terms of unions of increasing subsequences of the permutation $w=a_1\cdots a_n$. Thus we can build up the insertion tableau $P(w)$ one step at a time by applying Greene's theorem to the subpermutations (subsequences) of $w$ consisting of the numbers $1,2,\dots,i$. We can similarly compute the recording tableau $Q(w)$ since $Q(w)=P(w^{-1})$. – Richard Stanley Feb 8 at 17:13

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.