TITLE: RSK and crystal operators
QUESTION [6 upvotes]: Is there a good reference on how RSK (and the 3 other variants)
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 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.

REPLY [2 votes]: I figured out the details, and wrote it up here. 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.
We manage to give a Schur-expansion for a class of LLT polynomials where the expansion was previously unknown.