Skew RSK dynamics: Greene invariants, affine crystals and applications to<i>q</i>-Whittaker polynomials

<jats:title>Abstract</jats:title><jats:p>Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late 1980s, we define deterministic dynamics on the space of pairs of skew Young tableaux<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508623000239_inline1.png" /><jats:tex-math>$(P,Q)$</jats:tex-math></jats:alternatives></jats:inline-formula>. We find that these skew RSK dynamics display conservation laws which, in the picture of Viennot’s shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508623000239_inline2.png" /><jats:tex-math>$0$</jats:tex-math></jats:alternatives></jats:inline-formula>-th Kashiwara operators reveals that the skew RSK dynamics possess symmetries induced by an affine bicrystal structure, which, combined with connectedness properties of Demazure crystals, leads to the linearization of the time evolution. Studying asymptotic evolution of the dynamics started from a pair of skew tableaux<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508623000239_inline3.png" /><jats:tex-math>$(P,Q)$</jats:tex-math></jats:alternatives></jats:inline-formula>, we discover a new bijection<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508623000239_inline4.png" /><jats:tex-math>$\Upsilon : (P,Q) \mapsto (V,W; \kappa , \nu )$</jats:tex-math></jats:alternatives></jats:inline-formula>. Here,<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508623000239_inline5.png" /><jats:tex-math>$(V,W)$</jats:tex-math></jats:alternatives></jats:inline-formula>is a pair of vertically strict tableaux, that is, column strict fillings of Young diagrams with no condition on rows, with the shape prescribed by the Greene invariant,<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508623000239_inline6.png" /><jats:tex-math>$\kappa $</jats:tex-math></jats:alternatives></jats:inline-formula>is an array of nonnegative weights and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508623000239_inline7.png" /><jats:tex-math>$\nu $</jats:tex-math></jats:alternatives></jats:inline-formula>is a partition. An application of this construction is the first bijective proof of Cauchy and Littlewood identities involving<jats:italic>q</jats:italic>-Whittaker polynomials. New identities relating sums of<jats:italic>q</jats:italic>-Whittaker and Schur polynomials are also presented.</jats:p>