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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.

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There is a detailed analysis in Chapters 7 and 8 of Bump and Schilling's Crystal Bases. They work through the connection between RSK and crystals in careful detail, though I don't recall how much detail they give on the variant you are interested in.

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  • $\begingroup$ Ah, yes I have been trying to find the relevant chapters of that book, but it does not seem to be available at my local library. Perhaps someone has the relevant pages scanned and willing to email me? $\endgroup$
    – Per Alexandersson
    Commented May 23, 2019 at 6:47
  • $\begingroup$ I bought the book, but the interaction with dual RSK is not there (it is the double crystal structure in this case I am interested in). $\endgroup$
    – Per Alexandersson
    Commented Jul 15, 2019 at 19:33
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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.

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