Key expansion of the flagged refined skew stable Grothendieck polynomial

The flagged refined stable Grothendieck polynomials of skew shapes generalize several polynomials like stable Grothendieck polynomials, flagged skew Schur polynomials. In this paper, we provide a combinatorial expansion of the flagged refined skew stable Grothendieck polynomial in terms of key polynomials. We present this expansion by imposing a Demazure crystal structure on the set of flagged semi-standard set-valued tableaux of a given skew shape and a flag. We also provide expansions of the row-refined stable Grothendieck polynomials and refined dual stable Grothendieck polynomials and the Schur P-functions in terms of stable Grothendieck polynomials $G_{\lambda }$ and in terms of dual stable Grothendieck polynomials $g_{\lambda }$.