# About $K$-rectification of increasing tableaux

Let $$T$$ be a standard Young tableaux on $$[n]$$. Denote the RSK algorithm $$\text{RSK}(w)=(P(T),Q(T))$$ for $$w\in\mathfrak{S}_n$$, where $$P(T)$$ is the Schencted insertion tableaux.

For $$1\leq i\leq j\leq n$$. Let $$T_{[i,j]}$$ be the skew SYT by restricting $$T$$ to the segament $$[i,j]$$. For a skew shape $$Y$$, define the rectification of Y, $$\text{Rect}(Y)$$ to be applying jeu de taquin on $$Y$$ to obtain a standard shape. See section 2.1 of this paper, in which $$\text{Rect}$$ is denoted as $$\text{std}$$.

It is well known that

For $$w\in \mathfrak{S}_n$$, $$T\in \text{SYT}_n$$. If $$P(w)=T$$, then $$\text{Rect}(T_{[i,j]})=P(w_{[i,j]})$$ for all $$[i,j]\subseteq [n]$$, where $$w_{[i,j]}$$ means restricting the permutation to the subalphabet $$[i,j]$$, e.g. $$126534_{[2,5]}=2534$$.

My question is: is there a $$K$$-theoretic analog of this property, in terms of Hecke insertion, $$K$$-jeu-de-taquin of increasing tableaux?

Specifically, define $$K$$-rectification by replacing jdt with $$K$$-jdt, and denote $$K$$-$$P(w)$$ the Hecke-insertion tableau of the word $$w$$. We say a Tableau is unique-rectification-target if it's unique in its $$K$$-Knuth class. (Definition 3.15 of [BS13], see also section 4 of [PP14])

Let $$T$$ be an increasing Tableau (of alphabet $$[n]$$) and $$Y=T_{[1,i]}$$ such that $$Y$$ is an SYT and $$1\cdots i\notin T\backslash Y$$ (so that there is no ambiguity). If $$K\text{-Rect}(T\backslash Y)$$ is a URT, then is it always true that: $$K\text{-Rect}(T\backslash Y)=K\text{-}P(w_{[i+1,n]})$$? where $$w$$ is the row-reading word of $$T$$, or even all words such that $$K\text{-}P(w)=T$$.

In general, is it true that $$\{K\text{-Rect}(T\backslash Y)\}=\{K\text{-}P(w_{[i+1,n]}):K\text{-}P(w)=T\}$$

For example, Let $$T=\begin{matrix}1&2&4\\3&5&6\\4&6&9 \end{matrix},Y=\begin{matrix}1&2\\3&\end{matrix}$$. We have $$K\text{-Rect}\left(\begin{matrix}*&*&4\\*&5&6\\4&6&9 \end{matrix} \right)=\begin{matrix}4&5&6\\5&9&\\6&& \end{matrix}$$ The row reading word of $$T$$ is $$w=\mathfrak{row}(T)=469356124$$, and $$K\text{-}P(w_{[4,9]})=K\text{-}P(469564)= \begin{matrix}4&5&6\\5&9&\\6&& \end{matrix}$$

Thanks !

• Nice question ;) – Sam Hopkins Jun 16 '19 at 21:40
• But one issue is that $K$-rectification is not necessarily uniquely defined because $K$-jdt lacks the same confluence properties as usual jdt. – Sam Hopkins Jun 16 '19 at 21:46
• Perhaps there is a nice interplay with K-theoretic crystals? de.arxiv.org/pdf/1904.09674.pdf RSK and crystal operators in the classical case are closely related... – Per Alexandersson Jun 17 '19 at 9:19
• @PerAlexandersson Thanks! – Sylvester W. Zhang Jun 17 '19 at 20:17
• @SamHopkins Actually I was very wrong. $K$-Rect is not uniquely defined. See the edit of my question. – Sylvester W. Zhang Jun 18 '19 at 1:16

$$K\text{-rect}\left(\begin{matrix}*&*&1&2\\*&1&3&4\\2&3\end{matrix}\right)=\begin{matrix}1&3&2\\2&4\end{matrix}$$ $$K\text{-}P(2313412)=\begin{matrix}1&2&4\\2&3\end{matrix}$$ I think the reason is that in the standard setting, jdt preserves Knuth equivalence and Knuth equivalence $$\iff$$Schensted insertion equivalence. But in $$K$$-theoretic version, $$K$$-jdt preserves $$K$$-knuth equivalence but $$K$$-Knuth equivalence if (but not only if) Hecke insertion equivalence.