Demazure crystals for flagged set-valued reverse plane partitions

In this paper, we build a Demazure crystal structure on the set of all flagged set-valued reverse plane partitions of a given skew shape $\lambda \mathrm{/}\mu$ and flag $\mathrm{\Phi }$. Our construction provides a unified generalization of the Demazure crystal structures previously known for the set of all flagged semi-standard set-valued tableaux and the set of all flagged reverse plane partitions. Consequently, we obtain a combinatorial expansion for the flagged hybrid Grothendieck polynomial $H_{\lambda \mathrm{/}\mu }({\mathbf{x}}_{\mathrm{\Phi }};\mathbf{t};\mathbf{w})$, introduced by Guo–Kang–Liu, in terms of key polynomials. Applying this expansion, we express the hybrid Grothendieck polynomial $H_{\lambda \mathrm{/}\mu }(\mathbf{x};\mathbf{t};\mathbf{w})$ in terms of both the stable Grothendieck polynomials $G_{\nu }(\mathbf{x})$ and the dual stable Grothendieck polynomials $g_{\nu }(\mathbf{x})$.