Inhomogeneous $q$-Whittaker Polynomials I: Duality and Expansions
Ajeeth Gunna, Michael Wheeler, Paul Zinn-Justin
TL;DR
The paper develops a unifying two-parameter family 𝔊^{(u,v)}_{λ} of symmetric polynomials arising from exactly solvable lattice models tied to 𝕌_q(sl_2[z^±]). By deriving Cauchy identities via the Yang–Baxter equation and introducing dual Hall–Littlewood–type objects 𝔍^{(u,v)}_{λ/μ}, it simultaneously encompasses q-Whittaker, inhomogeneous q-Whittaker, Grothendieck, and dual Grothendieck polynomials. A central achievement is a general expansion framework that expresses q-Whittaker and inhomogeneous variants in terms of each other and their duals, with explicit, positive combinatorial coefficients given by partition-function weights. The work illuminates duality structures and inversion relations in this broad family, and provides concrete branching formulas and degenerations to classical bases (Schur, Hall–Littlewood, Grothendieck). Overall, it lays a robust lattice-model foundation for systematically deriving Cauchy identities and positive expansions among central symmetric-polynomial families in a unified setting.
Abstract
We introduce a new family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_λ$ arising from exactly solvable lattice models associated with the quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$. The polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_λ$ 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.