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

TL;DR

This paper introduces a new family of symmetric polynomials derived from solvable lattice models, unifying several known polynomial families and providing identities and formulas via the Yang--Baxter equation.

Contribution

It presents a novel family of symmetric polynomials that generalize and unify existing polynomials, with derivations of identities and combinatorial formulas using integrable model techniques.

Findings

Unification of q-Whittaker, inhomogeneous q-Whittaker, and Grothendieck polynomials.

Derivation of Cauchy identities for the new polynomials.

Development of combinatorial formulas for transition coefficients.

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.