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](https://reader.elsevier.com/reader/sd/pii/S0097316505002013?token=B3981B70322E704ABFB6C25E0CE2F63F92CD0340D50644A85E67EB3EE647C4B415B5B1E042CE5E5FE2A08210F3341AA8), 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. $126534\_{[2,5]}=2534$.
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> 
> 


My question is: is there a $K$-theoretic analog of this property, in terms of [Hecke insertion](https://arxiv.org/abs/math/0601514), [$K$-jeu-de-taquin](https://arxiv.org/abs/0705.2915) of increasing tableaux?




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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]](https://arxiv.org/pdf/1306.5419.pdf), see also section 4 of [[PP14]](https://arxiv.org/abs/1404.4340))



> 
> 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 !
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About $K$-rectification of increasing tableaux
The statement is not true for increasing tableau.


$$K\text{-rect}\left(\begin{matrix}\*&\*&1&2\\\*&1&3&4\\2&3\end{matrix}\right)=\begin{matrix}1&3&2\\2&4\end{matrix}$$
$$K\text{-}P(2313412)=\begin{matrix}1&2&4\\2&3\end{matrix}$$
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.


