Is there a good reference on how RSK [(and the 3 other variants)](https://arxiv.org/pdf/math/0510676.pdf)
interact with crystal operators on the semi-standard tableaux $(P,Q)$ in the image?


That is, we have biwords, $W$ which are in bijection with pairs of semi-standard tableaux $(P,Q)$ under RSK. Now, we act on $P$ or $Q$ with crystal raising/lowering operators $e\_i$ and $f\_i$.
These actions are defined on SSYTs. 
What happens with $W$? [Theorem 2.2 here](https://pdfs.semanticscholar.org/362d/ad8936a0521890cc4dd7efd851fdcf02f74a.pdf) almost answers my question (although it just references a paper by Lascoux that is a bit hard to parse), but I would like to know if there is a survey/book that goes into depth on this - in particular when considering the other variants of RSK.


I am in particular interested in variant III in Christian's survey. 
[For a quick reference, I describe this variant on my page here, including an example.](https://www.math.upenn.edu/~peal/polynomials/rsk.htm#rskPrime)



RSK and crystal operators
I figured out the details, and [wrote it up here](https://www.math.upenn.edu/%7Epeal/polynomials/tableauOperators.htm#crystals-RSK). I did not manage to find a good reference. There are a few nice surveys on RSK and on crystals, but a survey covering how different tableau operators interact would be nice to see someone type up.


For the interested, these properties above were needed for this project, where we look at a [type of skew q-Whittaker functions](https://arxiv.org/abs/1908.00083).
We manage to give a Schur-expansion for a class of LLT polynomials where the expansion was previously unknown.


