Hybrid Grothendieck polynomials

For a skew shape $\lambda \mathrm{/}\mu$, we define the hybrid Grothendieck polynomial $$G_{\lambda \mathrm{/}\mu }(\mathbf{\text{x}};\mathbf{\text{t}};\mathbf{\text{w}})=\sum _{T\in \mathrm{S}\mathrm{V}\mathrm{R}\mathrm{P}\mathrm{P}(\lambda \mathrm{/}\mu )}{\mathbf{\text{x}}}^{\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}(T)}{\mathbf{\text{t}}}^{\mathrm{c}\mathrm{e}\mathrm{q}(T)}{\mathbf{\text{w}}}^{\mathrm{e}\mathrm{x}(T)}$$ as a weight generating function over set-valued reverse plane partitions of shape $\lambda \mathrm{/}\mu$. It specializes to \begin{itemize} \item[(1)] the refined stable Grothendieck polynomial introduced by Chan–Pflueger by setting all $t_i=0$; \item[(2)] the refined dual stable Grothendieck polynomial introduced by Galashin–Grinberg–Liu by setting all $w_i=0$. \end{itemize} We show that $G_{\lambda \mathrm{/}\mu }(\mathbf{\text{x}};\mathbf{\text{t}};\mathbf{\text{w}})$ is symmetric in the $\mathbf{\text{x}}$ variables. By building a crystal structure on set-valued reverse plane partitions, we obtain the expansion of $G_{\lambda \mathrm{/}\mu }(\mathbf{\text{x}};\mathbf{\text{t}};\mathbf{\text{w}})$ in the basis of Schur functions, extending previous work by Monical–Pechenik–Scrimshaw and Galashin. Based on the Schur expansion, we deduce that hybrid Grothendieck polynomials of straight shapes have saturated Newton polytopes. Finally, using Fomin–Greene’s theory on noncommutative Schur functions, we give a combinatorial formula for the image of $G_{\lambda \mathrm{/}\mu }(\mathbf{\text{x}};\mathbf{\text{t}};\mathbf{\text{w}})$ (in the case $t_i=\alpha$ and $w_i=\beta$) under the omega involution on symmetric functions. The formula unifies the structures of weak set-valued tableaux and valued-set tableaux introduced by Lam–Pylyavskyy. Several problems and conjectures are motivated and discussed.