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

Abstract

We introduce a new family of symmetric polynomials ${\mathfrak{G}}_{\hat{I}}^{(\mathbf{u},\mathbf{v})}»$ arising from exactly solvable lattice models associated with the quantised loop algebra ${\mathcal{U}}_q({\mathfrak{sl}}_2[z^{\pm }])$. The polynomials ${\mathfrak{G}}_{\hat{I}}^{(\mathbf{u},\mathbf{v})}»$ 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.