Inhomogeneous $q$-Whittaker Polynomials I: Duality and Expansions

We introduce a new family of symmetric polynomials ${\mathfrak{G}}_{\lambda }^{(\mathbf{u},\mathbf{v})}$ arising from exactly solvable lattice models associated with the quantised loop algebra ${\mathcal{U}}_q({\mathfrak{s}\mathfrak{l}}_2[z^{\pm }])$. The polynomials ${\mathfrak{G}}_{\lambda }^{(\mathbf{u},\mathbf{v})}$ unify $q$-Whittaker polynomials, inhomogeneous $q$-Whittaker polynomials, Grothendieck polynomials and their duals. Using Yang–Baxter equation, we derive Cauchy identities and combinatorial formulas for the transition coefficients.