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

Abstract

We introduce a new family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ arising from exactly solvable lattice models associated with the quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$. The polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ 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.